8 Practice Reciprocal Trigonometric Functions: Fill & Download for Free

Thank you for asking - the way I would structure my (sketch* of a) proof in terms of the machinery of real analysis, in order to make it accessible to as wide an audience as possible and in order to make it as elementary as possible, is as follows.Proof: let [math]S(a)[/math] be a real, as we are told, to real function such that:[math]\displaystyle S(a)=\int_\limits{0}^{+\infty}\sin\left(x^a\right)\,\mathrm dx \tag{1}[/math]Our integrand, then, practically begs for the following substitution:[math]x^a = t \tag{2}[/math]from where it follows that:[math]ax^{a-1}\,\mathrm dx = \mathrm dt \tag*{}[/math]from where, with (2) and after multiplying both sides of the above relation by a single copy of [math]x[/math], it follows that:[math]at\,\mathrm dx = t^{\frac{1}{a}}\mathrm dt \tag{3}[/math]which is safe since we are assured that it is the case that [math]a>1[/math].For the sake of brevity let us agree on the following lightweight substitution that amounts to just a syntactic sugar, do not read much into it:[math]1 - \dfrac{1}{a} = s \tag{4}[/math]Hence, [math]s[/math] honors the following constraint:[math]0 < s < 1 \tag{5}[/math]because, again, we are told that [math]a>1[/math].Thus, our integral [math]S(a)[/math] becomes:[math]\displaystyle S(a)=\dfrac{1}{a}\;\int_\limits{0}^{+\infty}\dfrac{\sin t}{t^s}\,\mathrm dt \tag{6}[/math]and we shall not pollute our name space with [math]S(a,s)[/math] - with (4) it is abundantly clear what is what.For the readers who are not familiar with complex numbers in general the following maneuver may seem puzzling, mysterious and artificial but, in its essence, it is not. The moment we see an object like the one in (6) we, of course, know that the trigonometric function of sine is just one imaginary half of the whole complex story because its sibling, the trigonometric function of cosine, is never too far away.Moreover, the readers who are following along and who have heard enough of our preaching already know what is coming next - an abstraction:[math]\displaystyle C(a,b) = \dfrac{1}{a}\;\int_\limits{0}^{+\infty}\dfrac{\cos bt}{t^s}\,\mathrm dt \tag{7}[/math]where [math]b[/math] is a real number such that [math]b > 0[/math].Consequently, we may broaden our integral a bit via:[math]\displaystyle S(a,b)=\dfrac{1}{a}\;\int_\limits{0}^{+\infty}\dfrac{\sin bt}{t^s}\,\mathrm dt \tag{8}[/math]for which the constraint on [math]s[/math] my be relaxed somewhat: [math]0<s<2[/math], which, of course, always holds in this particular problem.Let us work with the integral [math]C(a, b)[/math] first - the integral [math]S(a,b)[/math] is dealt with in exactly the same fashion.It was the French mathematician Legendre (1752–1833) who suggested to name the following integral as the Eulerian integral of the second kind:[math]\displaystyle \Gamma(z) = \int_\limits{0}^{+\infty}x^{z-1}e^{-x}\,\mathrm dx \tag{9}[/math]which is known to modern students as the gamma function, in which the real part of the complex number [math]z[/math] must be strictly positive.Which integral do you think Legendre suggested to name as the Eulerian integral of the first kind? Correct - the beta function.Introduce the following substitution in (9):[math]x = ty,\;t>0 \tag{10}[/math]after which the integral in (9) can be rewritten as:[math]\displaystyle \dfrac{\Gamma(z)}{t^z} = \int_\limits{0}^{+\infty}y^{z-1}e^{-ty}\,\mathrm dy \tag{11}[/math]or, even more suggestively, as:[math]\displaystyle \dfrac{1}{t^z} = \dfrac{1}{\Gamma(z)}\, \int_\limits{0}^{+\infty}y^{z-1}e^{-ty}\,\mathrm dy \tag{12}[/math]Now take a look at our integrals [math]C(a, b)[/math] and [math]S(a, b)[/math] and then take a look at the relation in (12).What do you see?Take your time … very good.You see that the commonality between (12) and (7, 8) is the reciprocal of the variable/parameter [math]t[/math] raised to the power of [math]z[/math] in first case and the power of [math]s[/math] - in the second. That is fine because [math]s[/math] is a real number that satisfies the constraint in (5) and, thus, honors the requirement imposed by the definition of the gamma function that the real part of [math]z[/math] must be a strictly positive number:[math]\displaystyle \dfrac{1}{t^s} = \dfrac{1}{\Gamma(s)}\, \int_\limits{0}^{+\infty}y^{s-1}e^{-ty}\,\mathrm dy \tag{13}[/math]Hence, put the RHS image of (13) into the RHS of (7) in place of [math]t^{-s}[/math]:[math]\displaystyle C(a,b) = \dfrac{1}{a\Gamma(s)}\;\int_\limits{0}^{+\infty}\cos bt\,\mathrm dt \;\int_\limits{0}^{+\infty}y^{s-1}e^{-ty}\,\mathrm dy \tag{14}[/math]Next, we do not forget to carefully justify the right to switch the order of integration in (14). This is left to the reader as an exercise:[math]\displaystyle C(a,b) = \dfrac{1}{a\Gamma(s)}\;\int_\limits{0}^{+\infty}y^{s-1}\,\mathrm dy \;\int_\limits{0}^{+\infty}e^{-ty}\cos bt\,\mathrm dt \tag{15}[/math]where the innermost integral in (15) has been computed in its essence on multiple occasions here on Quora.It just so happened that we used it in my previous answer that dealt with the relevant Fourier series:[math]\displaystyle \int e^{ax}\cos bx\;\mathrm dx = e^{ax}\dfrac{a\cos bx [/math][math][/math][math]+ b\sin bx}{a^2+b^2} [/math][math][/math][math]+ C_c \tag*{}[/math][math]\displaystyle \int e^{ax}\sin bx\;\mathrm dx = e^{ax}\dfrac{-b\cos bx [/math][math][/math][math]+ a\sin bx}{a^2+b^2} [/math][math][/math][math]+ C_s \tag*{}[/math]as was shown in this Quora answer.Thus:[math]\displaystyle \int_\limits{0}^{+\infty}e^{-ty}\cos bt\,\mathrm dt = \dfrac{y}{y^2 [/math][math][/math][math]+ b^2} \tag*{}[/math]because the role of the parameter [math]a[/math] in the above indefinite integrals is played locally by the magnitude [math]-y[/math]. Since you are a diligent student of mathematics, you will not be lazy and write out the above result in terms of the relevant limits.Therefore, our integral [math]C(a,b)[/math] becomes:[math]\displaystyle C(a,b) = \dfrac{1}{a\Gamma(s)}\;\int_\limits{0}^{+\infty}\dfrac{y^s}{y^2 [/math][math][/math][math]+ b^2}\,\mathrm dy \tag*{}[/math]in which, after a fairly obvious substitution[math]y^2 = b^2 t \tag*{}[/math]we readily recognize the beta function which can be given a variety of definitions:[math]\displaystyle C(a,b) = \dfrac{b^{s-1}}{2a\Gamma(s)}\;\int_\limits{0}^{+\infty}\dfrac{t^{\frac{s-1}{2}}}{1+t} \,\mathrm dt \tag{16}[/math]For example, in the following popular way to define the beta function:[math]\displaystyle B(a,b) = \int_\limits{0}^{1}x^{a-1}(1-x)^{b-1}\,\mathrm dx \tag*{}[/math]with [math]a,b>0[/math] substitute[math]x = \dfrac{y}{1 [/math][math][/math][math]+ y} \tag*{}[/math]to obtain an alternative, and sometimes more convenient, definition of the same beta function:[math]\displaystyle B(a,b) = \int_\limits{0}^{+\infty}\dfrac{y^{a-1}}{(1+y)^{a+b}}\,\mathrm dy \tag*{}[/math]Putting [math]b = 1 - a[/math] in the above relation, we find:[math]\displaystyle B(a,1 - a) = \int_\limits{0}^{+\infty}\dfrac{y^{a-1}}{1+y}\,\mathrm dy \tag*{}[/math]which is an integral that can computed in a variety of ways. We omit the details here for the sake of brevity and show just the result:[math]\displaystyle B(a,1 - a) = \int_\limits{0}^{+\infty}\dfrac{y^{a-1}}{1+y}\,\mathrm dy = \dfrac{\pi}{\sin a\pi},\;0<a<1 \tag*{}[/math]and this is exactly what we have in (16):[math]\displaystyle C(a,b) = \dfrac{b^{s-1}}{2a\Gamma(s)}\cdot B\left(\dfrac{s+1}{2}, \dfrac{1-s}{2}\right) \tag*{}[/math]and, thus:[math]\displaystyle C(a,b) = \dfrac{b^{s-1}}{2a\Gamma(s)}\cdot \dfrac{\pi}{\sin \dfrac{s+1}{2}\pi} \tag*{}[/math]which, after the single [math]\pi/2[/math] shift of the argument of the trigonometric function of sine, is to say that[math]\displaystyle C(a,b) = \dfrac{1}{a}\;\int_\limits{0}^{+\infty}\dfrac{\cos bt}{t^s}\,\mathrm dt = \dfrac{b^{s-1}}{2a\Gamma(s)}\cdot \dfrac{\pi}{\cos \dfrac{s\pi}{2}} \tag{17}[/math]We have numbered the above result which seems generic enough to be reused and referenced at a future date.If we unwind the substitution in (4) then in terms of the parameter [math]a[/math] only we will have:[math]\displaystyle C(a,b) = \dfrac{b^{-\frac{1}{a}}}{2a\Gamma\left(1 - \dfrac{1}{a}\right)}\cdot \dfrac{\pi}{\cos \left(\dfrac{\pi}{2}\left(1-\dfrac{1}{a}\right)\right)} \tag*{}[/math]But we know that due to the Euler’s reflection formula:[math]\Gamma(z)\cdot\Gamma(1-z) = \dfrac{\pi}{\sin\pi z},\;z\not\in\mathbb{Z} \tag*{}[/math]Therefore:[math]\dfrac{1}{\Gamma\left(1 - \dfrac{1}{a}\right)} = \dfrac{\Gamma\left(\dfrac{1}{a}\right)\cdot\sin\dfrac{\pi}{a}}{\pi} \tag*{}[/math]and:[math]\displaystyle C(a,b) = \dfrac{b^{-\frac{1}{a}}}{2a\pi}\cdot \dfrac{\pi\cdot\Gamma\left(\dfrac{1}{a}\right)\cdot\sin\dfrac{\pi}{a}}{\sin \dfrac{\pi}{2a}} \tag*{}[/math]because, clearly, [math]\cos(\pi/2 - x) = \sin x[/math]. Canceling one copy of [math]\pi[/math] in the above numerator and denominator, we have:[math]\displaystyle C(a,b) = \dfrac{b^{-\frac{1}{a}}}{2a}\cdot \dfrac{\Gamma\left(\dfrac{1}{a}\right)\cdot\sin\dfrac{\pi}{a}}{\sin \dfrac{\pi}{2a}} \tag*{}[/math]Decompose the trigonometric function of sine in the above denominator via the double-angle identity:[math]\sin\dfrac{\pi}{a} = 2\sin\dfrac{\pi}{2a}\cos\dfrac{\pi}{2a} \tag*{}[/math]and cancel one copy of the resultant trigonometric function of sine:[math]\displaystyle C(a,b) = \dfrac{b^{-\frac{1}{a}}}{a}\cdot \Gamma\left(\dfrac{1}{a}\right)\cdot\cos\dfrac{\pi}{2a} \tag*{}[/math]Lastly, according to the multiplicative property of the gamma function:[math]\Gamma(z+1) = z\cdot\Gamma(z) \tag*{}[/math]Hence:[math]\Gamma\left(\dfrac{1}{a}\right) = a\cdot\Gamma\left(1+\dfrac{1}{a}\right) \tag*{}[/math]and:[math]\displaystyle C(a,b) = \dfrac{1}{a}\;\int_\limits{0}^{+\infty}\dfrac{\cos bt}{t^{1-\frac{1}{a}}}\,\mathrm dt = b^{-\frac{1}{a}}\Gamma\left(1+\dfrac{1}{a}\right)\cos\dfrac{\pi}{2a} \tag*{}[/math]Since in your particular case [math]b = 1[/math], we arrive at:[math]\displaystyle C(a)=\int_\limits{0}^{+\infty}\cos\left(x^a\right)\,\mathrm dx = \Gamma\left(1+\dfrac{1}{a}\right)\cos\dfrac{\pi}{2a} \tag{18}[/math]In exactly the same way we work with the integral [math]S(a, b)[/math] - the substitution for the reciprocal of [math]t^s[/math] from (13) will be the same.For the innermost integral we will have:[math]\displaystyle \int_\limits{0}^{+\infty}e^{-ty}\sin bt\,\mathrm dt = \dfrac{b}{y^2 [/math][math][/math][math]+ b^2} \tag*{}[/math]which means that the integral [math]S(a, b)[/math] will become:[math]\displaystyle S(a, b) = \dfrac{b}{a\Gamma(s)}\;\int_\limits{0}^{+\infty}\dfrac{y^{s-1}}{y^2 [/math][math][/math][math]+ b^2}\,\mathrm dy \tag*{}[/math]Use the same [math]y^2 = b^2t[/math] substitution and so on.You should be able to handle it from here on your own - note that this time, however, the power of the dummy integration variable [math]t[/math] (like in (16)) will be not [math](s-1)/2[/math] but rather [math](s-2)/2[/math]. Therefore, you should arrive at the following intermediate result similar to the one that we obtained in (17):[math]\displaystyle S(a,b) = \dfrac{1}{a}\;\int_\limits{0}^{+\infty}\dfrac{\sin bt}{t^s}\,\mathrm dt = \dfrac{b^{s-1}}{2a\Gamma(s)}\cdot \dfrac{\pi}{\sin \dfrac{s\pi}{2}} \tag{19}[/math]And again, unwinding the substitution in (4), using the Euler’s reflection formula, decomposing the trigonometric function of sine via the double-angle identity, using the multiplicative property of the gamma function and so on, we will have:[math]\displaystyle S(a,b)=\dfrac{1}{a}\;\int_\limits{0}^{+\infty}\dfrac{\sin bt}{t^{1-\frac{1}{a}}}\,\mathrm dt = b^{-\frac{1}{a}}\Gamma\left(1+\dfrac{1}{a}\right)\sin\dfrac{\pi}{2a} \tag*{}[/math]and since in your case [math]b = 1[/math], from the above relation it follows that:[math]\displaystyle S(a)=\int_\limits{0}^{+\infty}\sin\left(x^a\right)\,\mathrm dx = \Gamma\left(1+\dfrac{1}{a}\right)\sin\dfrac{\pi}{2a} \tag{20}[/math]what was required to prove.[math]\blacksquare[/math]*With a piece of chalk in your hand, in the format of a verbal mathematics exam, you may be able to present the above as a proof inside of 25 minutes or so and you may even get a decent grade (if you care about the grades). However, a rigorous proof is codified at such a low, atomic, level as to be consumable by a Turing Machine which, upon reading our candidate version, will spit out yay or nay promptly (after a finite number of steps/finite amount of time).The proofs codified in that way bring tears to my eyes because all the life, all the play of thought and all the improvisational tidbits that go into the construction of a proof are, basically, gone.In any case, once we have the results on the books, we can answer all sorts of derivative questions. For example, early on in the theory of improper integrals the following theorem is proved:if both integrals of the real-to-real functions [math]f(x)[/math] and [math]g(x)[/math] converge on the interval [math][a,+\infty)[/math] then the integral of their linear combination converges also:[math]\displaystyle \int_\limits{a}^{+\infty}\left(f(x)\pm g(x)\right)\,\mathrm dx = \int_\limits{a}^{+\infty}f(x)\,\mathrm dx \pm \int_\limits{a}^{+\infty}g(x)\,\mathrm dx\tag*{}[/math]Question: what is the value of the Pythegorean-identity-want-to-be:[math]\displaystyle \int_\limits{0}^{+\infty}\left(\cos\left(x^2\right) [/math][math][/math][math]+ \sin\left(x^2\right)\right)\,\mathrm dx \tag*{}[/math]where the squares jump off of the trigonometric functions themselves and hop onto their respective arguments?Well, we know:[math]\dfrac{\sqrt{2\pi}}{2} \tag*{}[/math]etc.

These things are usually covered in, at least, a tandem - in complex analysis the corresponding functions are often defined in terms of series, as shown in this Quora answer - see the formulas (12) and (13) in it, but here we shall cover the subject the old-fashioned way.The hyperbolic cosine, denoted [math]\cosh x[/math], may be defined symbolically as follows:[math]y = \cosh x = \dfrac{e^x [/math][math][/math][math]+ e^{-x}}{2} \tag{1}[/math]with the entire set of real numbers [math]\mathbb{R}[/math] as its natural domain.The graph of the hyperbolic cosine in (1), shown below:has a number of interesting geometric properties and physical interpretations.For example, the arc length of the above graph can be linearized or straightened with Euclidean tools alone. In other words, the length of an arc of a graph of a hyperbolic cosine starting at the vertex [math]y = 1[/math] can be measured off on demand along a straight line with a straightedge and a compass only. Ask a separate question for the proof of that statement.Consequently, the hyperbolic sine, denoted [math]\sinh x[/math], may be defined symbolically as follows:[math]y = \sinh x = \dfrac{e^x - e^{-x}}{2} \tag{2}[/math]with the entire set of real numbers [math]\mathbb{R}[/math] as its natural domain and the graph shown below:Just by looking at the graph of a function we can normally make an accurate prediction about whether the corresponding inverse function will be single- or multi-valued:if any straight line parallel to the [math]x-[/math]axis cuts the graph of a function at exactly one point then the corresponding inverse function will be single-valuedIn the opposite direction:if there exists at least one straight line that cuts the graph of a function at two or more points then the corresponding inverse function will be multi-valuedExamining the graphs shown, by the above criterion, we conclude that the inverse hyperbolic cosine is a multi-valued function while the inverse hyperbolic sine is a single-valued function.To derive an explicit expressions for the inverse functions it is possible to swap the roles of [math]x[/math] and [math]y[/math] in the functions’ symbolic definitions and solve the resultant equation for [math]y[/math] in terms of [math]x[/math].For the inverse hyperbolic cosine: swap the roles of [math]x[/math] and [math]y[/math] in (1):[math]x = \dfrac{e^y [/math][math][/math][math]+ e^{-y}}{2} \tag*{}[/math]Multiply both sides of the result by [math]2[/math]:[math]2x = e^y [/math][math][/math][math]+ e^{-y} \tag*{}[/math]Subtract [math]2x[/math] from both sides:[math]e^y [/math][math][/math][math]+ e^{-y} - 2x = 0 \tag*{}[/math]Multiply by [math]e^y[/math] through:[math]e^{2y} - 2xe^y [/math][math][/math][math]+ 1 = 0 \tag*{}[/math]and solve the obtained equation, quadratic in [math]e^y[/math], for [math]e^y[/math], designating [math]e^y = z[/math] if need be:[math]z^2 - 2xz [/math][math][/math][math]+ 1 = 0 \tag*{}[/math]from where:[math]z_{1,2} = x \pm \sqrt{x^2 - 1} \tag*{}[/math]or:[math]e^y = x \pm \sqrt{x^2 - 1} \tag*{}[/math]and we keep both roots. Why? Because both roots fall within the natural domain [math]\mathbb{R}^{+}[/math] of the previously studied function [math]\log[/math] (in base [math]e[/math]) since the said roots are strictly positive:[math]y = \log\left(x \pm \sqrt{x^2 - 1}\right), \;x\geqslant 1 \tag{3}[/math]We, thus, see that the inverse hyperbolic cosine, in (3), is a multi-valued function that can be decomposed into two distinct single-valued branches each of which is defined for [math]x\geqslant 1[/math].Graphically, the branch [math]y_{+} = \log\left(x [/math][math][/math][math]+ \sqrt{x^2 - 1}\right)[/math] corresponds to the [math]y\geqslant 0[/math] horn located in the first quadrant, shown in blue below:while the branch [math]y_{-} = \log\left(x - \sqrt{x^2 - 1}\right)[/math] corresponds to the [math]y\leqslant 0[/math] horn located in the fourth quadrant, shown in red above.It also should make sense, since to obtain a graph of an inverse function we reflect the graph of the parent function in the straight line [math]y = x[/math].Likewise, for the inverse hyperbolic sine: swap the roles of [math]x[/math] and [math]y[/math] in (2):[math]x = \dfrac{e^y - e^{-y}}{2} \tag*{}[/math]Multiply both sides of the result by [math]2[/math]:[math]2x = e^y - e^{-y} \tag*{}[/math]Subtract [math]2x[/math] from both sides:[math]e^y - e^{-y} - 2x = 0 \tag*{}[/math]Multiply by [math]e^y[/math] through:[math]e^{2y} - 2xe^y - 1 = 0 \tag*{}[/math]and solve the obtained equation, quadratic in [math]e^y[/math], for [math]e^y[/math], designating [math]e^y = z[/math] if need be:[math]z^2 - 2xz - 1 = 0 \tag*{}[/math]from where:[math]z_{1,2} = x \pm \sqrt{x^2 [/math][math][/math][math]+ 1} \tag*{}[/math]or:[math]e^y = x [/math][math][/math][math]+ \sqrt{x^2 [/math][math][/math][math]+ 1} \tag*{}[/math]but this time we keep one, positive, root only. Why? Because the natural domain of the previously studied function [math]\log[/math] (in base [math]e[/math]) is [math]\mathbb{R}^{+}[/math] and the second root is negative:[math]y = \log\left(x [/math][math][/math][math]+ \sqrt{x^2 [/math][math][/math][math]+ 1}\right) \tag{4}[/math]We, thus, see that the inverse hyperbolic sine, in (4), is a single-valued function, as expected:and, again, the above graph is the reflection of the original in the straight line [math]y=x[/math].On your own, soonish, you should say all the right words: continuous, monotonically de/increasing, differentiable, etc. and, eventually, prove these things also.Extra for experts:why on Earth hyperbolic and co/sine, of all things, for something that seems to be a mix of distinctly exponential objects?While a fair amount of terminology that you see in the middle and high schools and the first few years in college, as was duly noted by Sridhar, is fossilized ignorance, this time, however, there is at least a technical explanation by analogy:the relationships that tie together the traditional trigonometric functions with the locations of the points on the circumference of a unit circle are very similar to the relationships that tie together the newly defined functions with the locations of the points on the graph of an equilateral hyperbola and its semi-major axis.We take it that the following traditional set up is familiar to all the readers - let [math]\theta[/math] be the unit circle’s central angle [math]AOP[/math] measured in radians and let [math]l[/math] be the length of the circular arc [math]AP[/math] shown in red:Since the radius of the generating circle is a unity, for the area [math]S[/math] of the circular sector [math]AOP[/math], shown in gray, we have:[math]S = \dfrac{1}{2}\cdot l \cdot 1 = \dfrac{l}{2} \tag*{}[/math]But [math]l = r\cdot \theta = 1\cdot \theta = \theta[/math]. Hence, [math]\theta = 2S[/math].In other words, instead of interpreting [math]\theta[/math] geometrically as a central angle, which is what we normally do, we may interpret [math]\theta[/math], numerically, as twice the area of a circular sector, while all the traditional relationships still hold:[math]|OC| = \cos\theta, [/math][math][/math][math]\; |PC| = \sin \theta \tag*{}[/math][math]|AB| = \tan\theta, [/math][math][/math][math]\; |DE| = \cot\theta \tag*{}[/math]and so on. The reason we emphasised circular is because when we switch gears we will consider hyperbolic sectors.We now transfer the above set up from the context of a unit circle into the context of an equilateral hyperbola:[math]x^2 - y^2 = 1 \tag{5}[/math]but this time we treat [math]\theta[/math] astwice the area of the hyperbolic sector [math]AOP[/math]shown cross-hatched below (to make the diagram readable we omit the lower, Southern, portion of the hyperbolic sector but make no mistake - it is there and due to symmetry the missing portion looks exactly like its Northern counterpart):By visual inspection we conclude that the area sought-after is equal to twice the difference between the area [math]S_{\triangle PCO}[/math] of the right, by construction, triangle [math]PCO[/math] and the area [math]S_{PCA}[/math] of the planar region trapped under the relevant portion of the graph of the equilateral parabola, above the [math]x-[/math]axis and to the left of the vertical straight line [math]PC[/math]:[math]\theta = 2\left(S_{\triangle PCO} - S_{PCA} \right) \tag*{}[/math]Since the coordinates of the point [math]P[/math] are [math](x, y)[/math], it follows that:[math]S_{\triangle PCO} = \dfrac{xy}{2} \tag*{}[/math]The area of the planar region [math]S_{PCA}[/math] can be computed via the definite Riemann integral (with (5)):[math]\displaystyle S_{PCA} = \int\limits_1^xy\;dx = \int\limits_1^x\sqrt{x^2 - 1}\;dx \tag*{}[/math]Now, for demonstration purposes, we shall play out a mock discovery process here - since we do not yet now about the existence of the hyperbolic functions, we can not use that particular - and popular - substitution.Thus, avoiding the vicious circle for the sake of the purity of an experiment, we make a substitution due to L. Euler (actually - just one from the family of three):[math]\sqrt{x^2 - 1} = t - x \tag{6}[/math]From where:[math]x = \dfrac{t^2 [/math][math][/math][math]+ 1}{2t} \tag{7}[/math]and[math]dx = \dfrac{t^2 - 1}{2t^2} [/math][math][/math][math]\; dt \tag*{}[/math]From (7):[math]x^2 - 1 = \dfrac{\left(t^2 - 1\right)^2}{(2t)^2} \tag*{}[/math]implying that[math]\sqrt{x^2 - 1} = \dfrac{t^2 - 1}{2t} \tag*{}[/math]which means that our indefinite, for now (in order to not mess with the boundaries), integral degenerates into:[math]\displaystyle \int \dfrac{t^2 - 1}{2t}\cdot \dfrac{t^2 - 1}{2t} \;dt \tag*{}[/math]Here we simply break up the whole into three pieces:[math]\displaystyle \dfrac{1}{4}\int t\;dt - \dfrac{1}{2}\int \dfrac{dt}{t} [/math][math][/math][math]+ \dfrac{1}{4}\int \dfrac{dt}{t^3} \tag*{}[/math]each of which is elementary integrable:[math]\dfrac{t^2}{8} - \dfrac{\log t}{2} - \dfrac{1}{8t^2} \tag{8}[/math]The reverse substitution of the middle term via (6) is straightforward:[math]\dfrac{\log t}{2} = \dfrac{1}{2}\cdot\log\left(x [/math][math][/math][math]+ \sqrt{x^2 - 1}\right) \tag*{}[/math]Also from (6) for [math]t^2[/math] we have:[math]t^2 = 2x^2 +2x\sqrt{x^2 - 1} - 1 \tag{9}[/math]From (9), for the reciprocal of [math]t^2[/math], after the multiplication of both the numerator and the denominator of the fraction by the corresponding conjugate [math]\left(2x^2 - 1\right) - 2x\sqrt{x^2 - 1}[/math], we find that the denominator, magically enough, evaporates into a unity (verify - just because I made all the calculations on paper preparing this answer does not mean that you shouldn’t):[math]\dfrac{1}{t^2} = 2x^2 [/math][math][/math][math] - 2x\sqrt{x^2 - 1} - 1 \tag*{}[/math]Thus, reverse substituting the first and the last terms in (8), we come to:[math]\dfrac{t^2}{8} - \dfrac{1}{8t^2} = \dfrac{1}{2}\cdot x\sqrt{x^2 - 1} \tag*{}[/math]and our definite integral, thus, amounts to:[math]\displaystyle S_{PCA} = \dfrac{1}{2}\cdot\left(xy - \log(x+y)\right) \tag*{}[/math]where, again, we have taken the square root from (5).For some more magic, how cool is it that the area of the right triangle also worked out to be [math]xy/2[/math]:[math]\theta = 2\left(\dfrac{xy}{2} - \dfrac{xy}{2} [/math][math][/math][math]+ \dfrac{\log(x+y)}{2} \right) \tag*{}[/math]Meaning that:[math]\theta = \log(x+y) \tag*{}[/math]meaning that:[math]x [/math][math][/math][math]+ y = e^{\theta} \tag{10}[/math]But reusing (5) we have:[math](x [/math][math][/math][math]+ y)(x - y) = 1 \tag{11}[/math]Right?Divide (11) by (10) and cancel out the common [math](x+y)[/math] term:[math]x - y = e^{-\theta} \tag{12}[/math]Between (10) and (12) exclude [math]y[/math] by adding (12) to (10) and solve the resultant equation for [math]x[/math] in terms of [math]\theta[/math]:[math]x = \dfrac{e^{\theta} [/math][math][/math][math]+ e^{-\theta}}{2} = \cosh\theta \tag{13}[/math]And, once (13) is on the books, from either (10) or (12) we find for [math]y[/math]:[math]y = \dfrac{e^{\theta} - e^{-\theta}}{2} = \sinh\theta \tag*{}[/math]But [math]x = |OC|[/math] which in the new speak is equal to [math]\cosh\theta[/math]. Compare the new result with [math]x = |OC|[/math] in the context of a unit circle.And [math]y = |PC|[/math] which in the new speak is equal to [math]\sinh\theta[/math]. Compare the new result with [math]y = |PC|[/math] in the context of a unit circle.We now have made a cool discovery and we may proceed to develop all the relevant properties of the hyperbolic functions that follow:For example, in the context of a unit circle the straight line [math]AB[/math] was a tangent to the unit circle. Here, the straight line [math]AB[/math] is a tangent to the equilateral hyperbola. You should be able to prove that [math]|BA| = \tanh\theta[/math], and that [math]|DE| = \coth\theta[/math] and that:[math]\cosh^2\theta - \sinh^2\theta = 1 \tag*{}[/math]and so on.For a practical example of the usage of the hyperbolic functions in, say, physics see this Quora answer where a material point slides without friction along a straight line which rotates about a fixed point in the horizontal plane at a constant angular velocity.

Engineering MechanicsConcept of measurement of mass, force, time and space, system of units, Fundamental & Derived units, conversion of units, required accuracy of resultsGeneral Principles of static’s, Vector addition, subtraction and products,Resultant of distributed (linear & non linear) force systems,General conditions of equilibrium of co-planer forces, Laws of triangle, parallelogram and polygon of forces,Types of Beams, supports and loads, simple cases of axial forces, shear forces and bending moment diagrams,Problem involving friction on flat surfacesGeometrical properties of plane areasWork, energy, power, impulse, momentum, conservation of momentum and energyRectilinear and curvilinear motions, Tangential and normal components of Acceleration, Simple harmonic motionEngineering Mechanics LabRoof TrussTo find the various forces in various parts of wall craneTo verify the line of polygon on various forcesTo find coefficient of friction between various materials on inclined planTo verify the principle of moment in the disc apparatusHelical blockTo draw a load efficiency curve for a screw jackBooks RecommendedEngineering Mechanics4th edition by Irving H. shames, Prentice HallAsk a Question about Engineering MechanicsEngineering Mechanics Questions AnswersView LecturesLectures Notes of Engineering Mechanics - Coming SoonEngineering DrawingIntroduction:Drawing Instrument & their use.Types of lines & letters.ScalesDimensions & their Types.Planning of a Sheet.Types of Engineering Drawings.Geometric Constructions & Engineering Curves, Parabola, Ellipse & Hyperbola.Projections:Concept of Projection.Types of Projections, Orthographic Projection.1st Angle & 3rd Angle ProjectionOrthographic Projection of Points, Simple lines, simple planes & simple solids.Three Views of an object.Sectional Views.Isometric Views from given orthographic Views.Free Sketches. Of solid objects & building drawings.Building Drawings.Building symbols.Types of building drawings, proposed drawing, submission drawing, Working drawing & completion drawing.Introduction to Bridge Drawings.Engineering DrawingDrawing Exercises:Planning of sheet & practicing lines & letters.Orthographic Projections of given model, Sectional Views.Steel Structure Connections detail.Preparation of Plan, Elevation & Section of Single story simple buildings.Isometric & other three dimensional ViewFree Hand Sketches.Books RecommendedEssentials of Drafting by James D. Bethune.Engineering Drawing by N.D. Butt.Ask a Question about Engineering Drawing & DraftingEngineering Drawing & Drafting Q's n AnswersView LecturesEngineering Mechanics, 4th edition by Irving H. shames, Prentice HallComputer Applications in Civil Engg. & CommunicationsPart – A: Computer ApplicationsIntroduction to Computer HardwareIntroduction to Operating SystemsIntroduction to Word ProcessorsIntroduction to Spread SheetsIntroduction to Presentation SoftwarePart – B: Communication SkillsBrain Storming ProcessAnalysis of TopicAudience AnalysisSequencing the Presentation MaterialDetailing of MaterialPreparing ConclusionQuestion & Answer SessionsPreparation of Presentation in SoftwarePresentation to AudienceParticipating in seminars and interviewsPresenting Conference PapersBooks RecommendedMastering OfficeXP, by Gini CortrteCommunication Skills ,University of PhoenixAsk Question about Civil SoftwaresCivil Engineeirng Softwares AnswersView LecturesEngineering MechanicsApplied Linear AlgebraVector Algebra:Introduction to scalars and vectors, Vectors in the plane, Scalar and vector products, Lines in R2, R 3 and planes, Spheres, Orthogonal projections, Perpendicular distance from a point to a line and a plane, Vector spaces, Subspaces, Linear combinations, Linearly dependent and Independent set of vectors, Spanning of a vector spaces, Bases of a vector spaces and its applications in engineering and Business.Matrix Algebra:Introduction to matrices, Matrix operations, Inverse Matrix, Rank of a Matrix, Echelon form of a Matrix and its applications in our daily life situation problems, i-e in line-communication as Air-lines, Telephone-lines, Connecting cities by roads.Determinants:Determinants and its properties, Inverse of a matrix, Rank of a matrix, linearly dependent and independent by determinants.Linear system of equations:Independent, Dependent and Inconsistent system of equations and its graphical representation, Trivial and non-trivial solutions of homogeneous system of linear equations and its applications as linear models in Business, Economics, Science, Electric Circuits and other branches of engineering. Solution of linear system of equations by determinants and its applications as Leontief input-output matrix of the economy, Coding and decoding theory.Linear Transformations:Reflection operators, Projection operators, Rotation operators, Shear in x and y directions, Dilation and Contraction.Eigenvalues and Eigenvectors:Eigenvalues and eigenvectors and its applications as deformation, Markov processes as Mass-transit problems, Forecasting of weather and to develop the solution of the system of differential equations for mechanical system/electrical system and civil engineering, especially in public health engineering problems.Books RecommendedDavid C Lay, Linear Algebra and its Application,2nd Edition, Addison-Wesley Publication Jan 1998Ask a Question about Civil EngineeringCivil Engineering Questions AnswersView LecturesEngineering Mechanics, 4th edition by Irving H. shames, Prentice HallElectrical TechnologyPower Systems layout, generation, transmission, distribution and utilization of electric powerElectrical Elements and circuitsElectrical current, voltage, power, energy, Ohm’s law, inductance, capacitance, Kirchoffs laws. Introduction to node voltage and loop current methods.Principle of House wiring and Industrial wiringDiode Transistor and simple rectifier circuit.Electrical know how related to experimental design instrumentations like corrosion rate measurements, strain gauges, LDT’s LVDT’s. etc.Books RecommendedTheraja, B.L. Electrical Technology,S. Chand (21st or 26th edition)Ask a Question about Civil EngineeringCivil Engineering Questions AnswersView LecturesEngineering MechanicsSemester - 2 Syllabus & CoursesMechanics of SolidsSimple stress and strainTypes of stresses and strainsLoad extension diagram for different materialsHooke’s law, Modulus of elasticityLateral and volumetric strain. Poisson’s ratioTemperature stresses and compound barsTheory of torsion of solid and hollow circular shaftsAdvanced cases of shear force and bending moment diagrams for statically determinate beams. Relationship between Load, Shear and Moment.Theory of simple bending, Neutral Axis, Resisting moments and section modulus.Shear stresses in mono-symmetric beamsDeflection of beams by double integration, moment area and conjugate beam methodsMechanics of Solids LabInvestigation of Hook’s law that is the proportional relation between force and stretching in elastic deformation,Determination of torsion and deflection,Measurement of forces on supports in statically determinate beam,Determination of shear forces in beams,Determination of bending moments in beams,Measurement of deflections in statically determinate beam,Measurement of strain in a barBend test steel bar;Yield/tensile strength of steel bar;Recommended BooksStrength of Materials by Andrew Pytel & Ferdinand L. SingerMechanics of Materials by E.P. PopovAsk a Question about Mechanics of SolidsMechanics of Solids Questions AnswersView LecturesMechanics of Solids - Strength of MaterialsComputer Programming for Civil EngineersIntroduction to Programming & Visual BasicVisual Basic Code ComponentsData TypesVariablesControl StructuresProceduresArraysFile I/O operationsLabVisual Basic Code Components (program examples)Data Types (program examples)Variables (program examples)Control Structures (program examples)Procedures (program examples)Arrays (program examples)File I/O operations (program examples)Programming related to Civil EngineeringMatrix addition, subtraction and multiplication.Beam analysis program for simply supported, cantilever, Overhanging BeamProgram for finding resultant of forces and its angleProgram for stress analysis of composite barRecommended BooksVisual Basic 6 by Deitel & Deitel, T.R. NietoVisual Home Page by Deitel & Deitel, T.R. NietoAsk a Question about Engineering MechanicsEngineering Mechanics Questions AnswersView LecturesVB.Net for Civil EngineeringEngineering MaterialsCements, Ceramics, and Refractories:Manufacture, properties, and application of lime, cement, ceramic and bricksMortars and concrete, Pozzolanic material, effects of various chemicals on cement and concrete. Methods of protection, strength and test of building stone. Quarrying and dressing of stone.Timbers:Varieties and uses of important timbers, method of seasoning and sawing Decay, seasoning and preservation of timber, laminated materials.Glass and Plastics:Composition, varieties, properties and use of glass, plastic, laminates and adhesive.Metals:Composition and properties of ferrous and non ferrous metals used in civil engineering. Effect of various heat treatments on the properties of steel and its alloys. Corrosion and methods of corrosion control. Properties of thermal insulation material for use in buildings.Paints and Varnishes:Composition, preparation, properties, test and uses of paints, plasters, varnishes and distempers.Other Materials:Acoustical material and geo-textiles, properties and uses of asphalt, bitumen, rubber and asbestos, laminates and adhesive.LabGradation of coarse aggregateGradation of fine aggregateFineness of cementSetting timeNormal ConsistencyInitial Setting timeFinal Setting timeTensile Strength of BriquetteCompressive strength of mortar cubeSoundness test of CementDensity of CementSlump TestRecommended BooksNeville A. Properties of Concrete, English Language Book SocietySmith R. C. Material of Construction, McGraw-HillAsk a Question about Engineering MaterialsEngineering Materials Questions AnswersView LecturesEngineering Materials LecturesCalculusSingle Variable Calculus:Basic concepts of single variable function, Continuous, discontinuous and piecewise continuous functions, Periodic, odd and even functions, algebraic functions, Transcendental functions and its graphical representations, Applications of functions in our daily life situations.Differential Calculus:Limits and continuity, Interpretation of a derivative, Geometric interpretation, Total differential and its applications in our daily life situations, The use of a table of different type derivatives, Higher order derivatives, Tangents and normals, Approximation of a function at a particular point by Taylor's and Maclaurin's series, Maximum and minimum values of a function, The first derivative test, The second derivative test, Point of inflexion and its applications in business and engineering.Integral Calculus:Basic concepts of integration, A table of integral formulas, Some rules of integration, Definite integrals, The area bounded by a curve, Integration by parts, Integration as the limit of a sum, Volume of revolution, and its applications in our daily life situationsMultivariate Calculus:Basic concepts of multivariate function, Level curves and surfaces, Limits and continuity, Partial differentiation, Geometric interpretation, higher partial derivatives. Tangent planes, Total differential, Vector functions and its differentiation and integration, The directional derivative, The gradient, Scalar and Vector fields, Normal property of the gradient, Divergence , Curl, Tangent planes and normal lines, Extrema of functions of two variables, Second partials test, Extreme value theorem, Method of Constrained optimization and Lagrange multipliers.Recommended BooksRobert Davison, Addison Wesley, Mathematics for EngineersAntom, H. Calculus and Analytic Geometry, Johney Wiley and Sons.Toff and Mckay, Practical MathematicsS.A.H.Rizvi, Engineering MathematicsAsk a Question about Civil EngineeringCivil Engineering Questions AnswersView LecturesEngineering MechanicsEngineering GeologyIntroduction to GeologyImportance of Geology for Civil Engineering ProjectsPhysical properties and identification of common rocks forming mineralsRocks formation and classification:According to the mode of occurrencesAccording to the Silica contentsWeather and erosion:Weather classification, fresh, slightly weathered, moderately weathered etc.Discontinuity classification:Joints, faults and other fractures, micro structural features, such as lamination, cleavages, foliations, spacing of discontinuities as close, wide, medium etc,Description of Rock masses as thickly bedded or thinly beddedIdentification of filling in joints, sand clay and breccias etcColor of grains, description with respect to the rock color and identification as a course grained, hardness classification as soft with respect to testGeological classification and identification of Rocks by geological namesIdentification and subordinate constitutions in rocks samples such as seams or branches of other types of minerals for example, Dolomite, Lime stone, Calcareous sand stone, sand.Classification of Durability of Rocks in Dry and wet condition with durability testEngineering and physical properties of rocksGeological technical properties of rocks used as building stones, as a decorated stones and as a industrial rocks such as color, luster, streak, specific gravity, water absorption and unit weight etc.Brief Introduction to structural Geology:Plate Tectonics with respect to the global application, earthquakes, causes of earthquakes, protective measures against earthquakes and zoning of earth quakes in PakistanRole of geology in selection of sites for dams, reservoirs, tunnels and other civil engineering structuresBrief introduction of local geologyMechanics of Solids 2Analysis of stresses and strains at a point due to combined effect of axial force, shear force and bending momentMohr’s circle for stresses and strainsRelationship between elastic constantsTheories of FailureUnsymmetrical BendingShear Stresses in mono-symmetric beamsShear stress distribution in unsymmetrical beamsShear flow, shear center, concentration of stressesThick and thin walled cylindersAnalysis of curved beams and beams on elastic foundationShort eccentrically loaded columnsCore of a sectionRankine Gordon formula for intermediate columns, slenderness ratioEccentrically Loaded ColumnsInelastic behavior of beams in flexureShape factor of a section.Recommended BooksStrength of Materials by Andrew Pytel & Ferdinand L. SingerMechanics of Materials by E.P. PopovAsk a Question about Mechanics of SolidsMechanics of Solids Questions AnswersView LecturesSolid Mechanics LecturesConstruction Engineering & Graphics - Building Construction & Design + AutoCADBuilding Construction:Site Selection for a building, Orientation and Setting out for constructionLayout Techniques:Layout Techniques with special reference to buildings. Excavation in different types of soilsMasonry constructionTypes of stone and brick masonry; bonds in brick masonry, Alignments, Plumbs, leveling & cambering.Form workForm work for general in-situ construction, props, bracing and horizontal shuttering platforms.Damp proofing in building, Interior and exterior surface finishesVarious types of floors & roofing systems, planner & non-planer roofing system, roof treatments.Expansion joints and construction jointsWood work in building constructionOther Engineering Projects.An over view of construction aspects of different types of engineering projects, e.g. buildings, retaining structures, bridgesConstruction specificationCommon defects in building construction their causes and remedial measuresLabArchitectural Drawings, Structural Drawings, Plumbing and Electrical Work.Computer Aided Drawing (AutoCAD)General and basic know how related to computer aided drafting, e.g., coordinate system, drawing setup procedure, basic draw commands, basic edit commands; Layers, creating test and defining styles options, block and drawing import/export options; Cross hatching, save and plot (2D) and isometric drawings. Use of Auto CAD in drawing plans, elevation and section of single and double-storey buildingsStructural AnalysisINTRODUCTION TO STRUCTURAL ANALYSIS:Definition of structure, types of structures: pin jointed and rigid jointed. Types of structural members. Types of beams, supports and loads. Stability of structures. Redundancy. Determinate and indeterminate structures, Degree of indeterminacy. Sign conventions for bending moment and shear force etc.ANALYSIS OF STATICALLY DETERMINATE RIGID JOINTED PLANE FRAMES:Definition. Analysis of determinate frames.ANALYSIS OF DETERMINATE PLANE TRUSSES:Definition and common types of trusses. Classification of co-planar trusses. Methods of analysis of trusses; Method of joints, Method of sections, Graphical method.ROTATIONS AND DEFLECTIONS:Deflection diagrams and elastic curves. Castigiano’s theorem for trusses beams and frames. Unit load method. Theorem of virtual work for trusses beams and frames.ARCHES:Definition. Linear arch. Eddy’s theorem. Three hinged parabolic and circular arch. Bending Moment and shear force diagrams. Influence lines for shear, thrust and moment.INFLUENCE LINES:Definition. Influence lines for statically determinate beams and paneled girders.Influence lines for reaction, shear and bending moment of statically determinate beams and paneled girders. Influence lines for axial forces in trusses. Influence lines for composite structures.TRAVELLING LOADS:Maximum bending moment and shear force at any section. Criterion for maximum moment and shear.Absolute maximum bending moment.CABLES AND SUSPENSION BRIDGES:Cables. Stiffened suspension bridges with three hinged stiffening girders.Shear force and bending moment diagrams.THREE MOMENT EQUATION:Derivation of 3-moment equation. Application of 3-moment equation to the analysis of indeterminate beamsLabsDemonstration of various types of structures and supports.Demonstrate the stability of structures using model structures.Determination of the horizontal thrust and maximum bending moment in a three hinged parabolic arches.Determination of the horizontal thrust and maximum bending moment in a two hinged parabolic arches.Determination of the deflections and rotations in overhanging beams.Demonstration of influence lineInvestigation of the buckling strutsDetermination of shear centeRecommended BooksWang, C.K. Intermediate Structural Analysis, McGraw HillWest, H.H. Analysis of Structures, John Wiley & Sons.Hibbeler, R.C. Structural Analysis, Prentice Hall.Ask a Question about Structural EngineeringStructural Analysis Questions AnswersView LecturesStructural Analysis LecturesTown Planning & ArchitectureArchitectureHistorical DevelopmentGeneral introduction to history of architecture; Emergence/Development of Islamic Architechure.InfluencesGeographical, climatic, religious, social, historical.PrinciplesTruth or purpose & beauty.QualitiesStrength, vitality, grace, breadth and scale.FactorsProportion, color and balance.Use of MaterialsStone, wood, metals, concrete, composites, ceramics.General Treatment to Plan of BuildingsWalls and their construction; Openings and their position, character and shape; Roofs and their development and employment; Columns and their position, form and decoration; Molding and their form decoration, Ornament as applied to any buildings.Town Planning DefinitionsTrends in urban growth; Objectives of town planning; Modern planning in Pakistan and abroad.Preliminary StudiesStudy of natural resources, economic resources, legal and administrative problems, civic surveys and preparation of relevant maps.Land Use PatternsVarious theories of land use pattern. Location of Parks and recreation facilities, public and semi-public buildings, civic centers, commercial centers, local shopping centers, public schools, industry & residential areas.Street PatternsLay out of street, road crossing & lighting; Community planningCity ExtensionsSub Urban development, Neighborhood Units, Satellite Town and Garden CityUrban PlanningIssues related to inner city urban design and emergence/up gradation of squatter settlements.Introduction to Urban & Regional Planning using GISRecommended BooksSnyter, J.C. Introduction to Urban Planning, Milwaukee McGraw Hill Book CoFletcher, S.B. A History of Architecture. The Athlone Press.Ask a Question about Town PlanningTown Planning Questions AnswersView LecturesTown Plannig Lectures - Coming SoonMechanical TechnologyBasics of ThermodynamicsThermodynamic systems,Laws of thermodynamics,Laws of perfect gasesEnergy equationInternal Energy, Enthalpy and entropy of the working fluidsPrime Movers:Internal combustion engines: type, working principle, cycle operation and performance,Steam EnginesSteam TurbinesAir-compressorsAir-Conditioning:Introduction to Air-conditioning and refrigeration. Heating and cooling load and its calculations, comfort chart, outline of AC systemsLabPractical # 1: To study the different components of petrol engine.Practical # 2: To study the cooling system of automobile engine.Practical # 3: To study the lubrication system of automobile engine.Practical # 4: To study the ignition system of automobile engine.Practical # 5: To study the fuel system of automobile engine.Practical # 6: To study the air-intake system of automobile engine.Practical # 7: To study the 2-Stroke Reciprocating Engine.Practical # 8: To study the vapor compression system.Practical # 9: To study different components of refrigeration and air-conditioning system.Practical # 10: The layout of boiler room.Practical # 11: To study the boiler of the steam engine power plant.Practical # 12: To study the steam Engine of the steam engine power plant.Practical # 13: To study the turbine of the steam engine power plant.Practical # 14: To study the condenser of steam engine power plant.Differential EquationsOrdinary Differential Equations:Basic concepts of ordinary differential equation, General and particular solutions, Initial and boundary conditions, Linear and nonlinear differential equations, Solution of first order differential equation by separable variables and its applications in our daily life situations, The techniques like change of variable, homogeneous, non-homogeneous, exact, non-exact, linear and nonlinear Bernoulli could be used in case of complications. Solution of second order differential equation by theory of operators and its applications as forced and free oscillations, The extension of second order solution criteria to higher order differential equations, Solution of the system of differential equations by theory of operators and its applications in our daily life situations.Partial Differential Equations:Basic concepts, Linear and nonlinear p.d.equations, Quasi linear and Quasi nonlinear p.d.equations, Homogeneous and non-homogeneous p.d.equations, Solutions of p.d.equations, Boundary and initial conditions as Dirichlet condition, Neumann condition, Robbins/Mixed condition, Classification of p.d.equations as Elliptic, Parabolic and Hyperbolic.Analytic solution by separation of variables of the Steady-State Two-Dimensional Heat equation/Laplace equation and Unsteady-State One-Dimensional Heat equation/Diffusion equation with homogeneous and nonhomogeneous boundary conditions. D'Alembert's solution of Two-Dimensional Wave equation with homogeneous and nonhomogeneous boundary conditions.Fourier Series:Periodic waveforms and their fourier representations, Calculating a fourier series, Fourier series of odd and even functions, Half range fourier series, Fourier series solution for the above p.d.equations.Recommended BooksKreyszig, E.Advanced Engineering Mathematics, Wayne and Erson.Abell & Braselton, Brooks /Cole, Modern Differential Equations, Second editionAsk a Question about Engineering MechanicsCivil Engineering Questions AnswersDownload Kreszig SolutionDifferential Equations SolutionsFluid MechanicsIntroduction to fluid Mechanics & its classification.History of fluid MechanicsApplication of Fluid mechanics in Civil Engineering.Distinction between solids and fluids.Physical Properties of Fluids:Density, Specific weight, Specific Volume, Specific gravity, Viscosity.Newton’s Law of viscosity. Surface tension. Compressibility of fluids.Fluid Statics:Static pressure, Pressure height relationship, absolute and gauge pressure,Measurement of Pressure, Barometer, Bourdon gauge, Pizometer tube, simple and differential manometer, Basic principal of various pressure measuring instruments.Forces on submerged plane and curved bodies. Buoyancy and Stability of submerged and floating bodies.Fluid Kinematics:Basics Concept about steady and unsteady flow, Laminar and Turbulent flow, Path lines, stream line, stream tube, uniform and non uniform flow.Basic Equations:Continuity Equation, Energy Equation and Momentum Equation.Application of Energy Equation and Continuity Equation to incompressible fluids.Fluid Measurement:Venturimeter, Orifices, Mouth pieces and Nozzles, pitot tube, Weirs and Notches.Flow through Pipes:Laminar Flow through pipes, Darcy’s Weisbach equation for flow through pipes,Hydraulic and Energy gradient lines, Losses in Pipe lines. Transmission of Energy through PipesFlow through Open Channel:Uniform flow through open channels. Chezy’s and Manning’s formula for uniform Flow through open channels. Most efficient cross section of open channel.LabsDemonstration of various parts of Hydraulic Bench.Experimental Study of laminar and turbulent Flow.Experimental Study of tube gauges and Dead weight pressure gauges.Calibration of Orifices by Various Methods.Calibration of venturimeter.Calibration of Rectangular and Triangular Notch.Verification of Bernoulli’s theorem.Determination of Metacentric height.Study of Various losses through Piping arrangements.Recommended BooksRobert L.Daughetn, Joseph B. Franzini, Fluid Mechanics with Engineering applicationsE.H.Lewitt, Hydraulics and Fluid MechanicsAsk a Question about Fluid MechanicsFluid Mechanics Questions AnswersView LecturesHydraulics & Fluid MechanicsSurveying & Levelling IIntroductionSurveying instruments; chains, tapes, steel bands, their types & uses.Chain SurveyingRanging & Chaining of Survey lines. Field work & Plotting of Chain surveyCompass SurveyingPrismatic Compass& Surveyor compassUses, Bearings, Local attraction, Fieldwork & Plotting.Plane Table SurveyingParts and accessories. Methods of surveying. Two-point and three-point problems.LevelingGeneral principle. Types of levels and their temporary and permanent adjustments. Methods of levelling. Reduction of levels, Precise levelling and Trigonometric Leveling.Theodolite.Types and uses of theodolites. Temporary and permanent adjustments. Measurement of horizontal and vertical angles.Tacheometrical surveyingMethods of tacheometrical surveying. Field work and computationTraversingTraversing with prismatic compass, theodolite and plane table, computations and adjustments of traverse, transformation of Co-ordinatesOmitted MeasurementsCalculation of Areas and VolumesEarthwork calculations. D.M.D. method, Simpson rule and trapezoidal rule.LabsPractice on measurement of distances and introduction to measuring instrumentsChain Surveying and plottingCompass TraversingPlane Table by methods of radiations and IntersectionsTwo Points ProblemThree Points ProblemLevel adjustments by two-peg method.Profile and Cross-SectioningTheodolite traversingRecommended BooksKavanash,B. Surveying principles and Application, Prentice HallIrvine, W. Surveying for Construction, McGraw Hill.Davis, R.E. Surveying Theory and Practice, McGraw HillAsk a Question about Surveying & LevellingEngineering Surveying Questions AnswersView LecturesEngineering SurveyingEngineering EconomicsBasic ConceptsIntroduction to Engineering EconomicsTime Value of MoneyDifferent Methods of Analysis (Present Worth Analysis, Annual Worth Analysis etc.)Concept of Internal Rate of Return & Its use in Engineering ProjectsVarious Types of Costs (Maintenance, Repair, Other Overhead Costs etc. ) & Benefits & Its Use in Engineering ProjectsEconomical Life of Construction EquipmentInflationDepreciation & Its TypesCost EscalationIntroduction to Break Even Analysis & Its Application to Engineering ProjectSensitivity AnalysisDemand and Supply AnalysisTheory of Pricing & Theory of Production and Laws of ReturnFunding, Funding Agencies and Planning CommissionTypes of Business OrganizationsLabor problems, Labor Organization, Prevention and Settlement of DisputesRecommended BooksDonald G. NewmanE.Peul DegarmoAsk a Question about Engineering EconomicsEngineering Economics Questions AnswersView LecturesEngineering Economics Lecture NotesStructural Analysis IIMethod of Consistent Deformation:Analysis of statically indeterminate beams, frames and trusses. Castigliano's second theorem (Method of least work).Slope Deflection MethodDerivation of equation, Analysis of continuous beams and portal frames without joint movement. Analysis of continuous beams and portal frames due to yielding of supports. Analysis of portal frames with joint movement. Analysis of closed frames and multistory portal frames.Moment Distribution MethodConcept, Distribution and carry over factors, Analysis of continuous beams and portal frames without joint movement. Analysis of continuous beams and portal frames due to yielding of supports. Analysis of continuous beams and portal frames with joint movement. Analysis of multistory portal frames.Plastic AnalysisElastic versus Plastic Analysis in Steel Design, Upper & Lower Bound Theorems, Statical Moment & Mechanism Methods. Design of Beams, Propped Cantilever, Fixed ended & Continuous beams & frames, Design ExamplesAnalysis of ArchesAnalysis of two hinged circular and parabolic arches with supports at different levels.Influence Lines for Indeterminate BeamsUse of Clark Maxwell and Betti’s Theorm, Law of Reciprocal Deflections, Influence lines for moment and reactions in proposed cantilevers. Influence lines for support reaction of two and three span continuous beams. Influence lines for Shear Force and Bending Moment at any section of the beam.Recommended BooksWang, C.K. Intermediate Structural Analysis, McGraw Hill.West, H.H. Analysis of Structures, John Wiley 7 Sons.Hibbeler, R.C. Structural Analysis, Prentice Hall.Ask a Question about Structural EngineeringStructural Analysis Questions AnswersView LecturesStructural Analysis LecturesNumerical AnalysisInterpolation and extrapolation;Finite differences, forward, backward and central differences and its operators form, Linear and higher order interpolating polynomials, Newton's Gregory forward & backward difference interpolation formulas and its utilization as extrapolation, Lagrange’s interpolation, Numerical differentiation based on differences,Numerical integration;Trapezoidal and Simpson's approximations, Romberg integration process,Numerical Solution of non-linear equations;Bracketing and iteration methods and its applications as multiple root methods,Direct solution of the system of linear equations;Gauss-elimination, Direct and indirect factorization, symmetric factorization, tri-diagonal factorization, Iterative methods like Jacobi's iteration and Gauss-Seidel iteration,Numerical solution of initial value problems;Single-Step methods and its comparison with Taylor's series expansion, Multi-Step methods, Higher order differential equations, System of differential equations,Numerical solution of linear and nonlinear boundary value problems.Soil MechanicsSoil FormationSoil and its constituents, weathering of rocks and types of soils. Description and Identification of soil (Visual-Manual Procedure), Mineralogy of soil solids.Physical PropertiesWater content, void ratio, porosity, degree of saturation, specific gravity, unit Weight and their determination, Atterberg limits, sieve analysis, hydrometer and Pipette analysis, Stoke’s law, grain size distribution.Soil Classification Grain size classification, Bureau of soils,. M.I.T., Unified, AASHTO and ASTM Classification systems. Textural Classification by triangular chart, united soil classification system, AASHTO soil classifications.Permeability and SeepageDefinition, Hydraulic gradient, Darcy's Law, Factors affecting permeability, Permeability of stratified soils, Laboratory and field determination of coefficient of permeability.Seepage force, quick sand condition, flow nets, boundary conditions, graphical method of flow net construction, determination of quantity of seepage, two dimensional flow, Laplace Equation, seepage through earth dams, design of filtersCompactionDefinition, Compaction fundamentals, Moisture density relationships,Standard Proctor test and modified AASHO test for compaction, Factors affecting compaction, Compaction equipment, properties and structure of compacted soils, Specifications, field control and measurement of in-situ density, CBR test.Vertical stresses in soilsDefinition, stresses caused by self weight of soil, Geostatic stresses, stresses caused by point loads and uniformly distributed loads: Boussinesq and Westerqaard theories, Pressure bulb, stress distribution diagram on horizontal and vertical planes; stress at a point outside loaded area, Newmark’s charts and 2:1 MethodSoil ExplorationImportance of soil exploration, soil exploration methods; probing, test trenches and pits, auger boring, wash boring, rotary drilling, Percussion drilling and geophysical methods, soil samples, Disturbed and undisturbed samples, In-situ tests (SPT, CPT and PLT)Introduction of related SoftwareLabsIdentification of Soil (Visual Manual Procedure)Determination of Moisture content of soilDetermination of specific gravity of soilDetermination of liquid limit of soilGrain-size analysis of soil (including both mechanical and hydrometer analysis)Determination of Plastic limit and Plasticity Index of soilDetermination of shrinkage limit of soilClassification of soil according to AASHTO and USCSModified/Proctor Compaction TestConstant Head Permeability test (Granular Soil)Falling Head Permeability (Granular and Fine grained soils)