What is $$\int_0^\infty e^{-t}\log tdt?$$  Plugging into  yields that it's $-\gamma$, where $\gamma$ is Euler's constant. How can we get it? Maybe by some contour integration?
87789
I'm trying to show that  $$\lim_{n \to \infty} \left[\sum_{k=1}^{n} \frac{1}{k} - \log n\right] = -\int_0^{\infty} e^{-t} \log t \,dt.$$  In other words, I'm trying to show that the above definitions of the Euler-Mascheroni constant $\gamma$ are equivalent.  In another post here (which I can't seem to find now) someone noted that  $$\int_0^{\infty} e^{-t} \log t \,dt = \left.\frac{d}{dx} \int_0^{\infty} t^x e^{-t} \,dt \right|_{x=0} = \Gamma&#39;(1) = \psi(1),$$  where $\psi$ is the digamma function.  This may be a good place to start on the right-hand side.  For the left-hand side I was tempted to represent the terms with integrals.  It is not hard to show that  $$\sum_{k=1}^{n} \frac{1}{k} = \int_0^1 \frac{1-x^n}{1-x} \,dx,$$  but I'm not sure this gets us anywhere.  Any help would be greatly appreciated.
49255
I read somewhere that   $$ \int_{0}^{\infty} \frac{x^3}{e^{x} - 1} = \frac{\pi^4}{15}$$  Does anyone see a way to prove this? My first idea was doing a contour integration and use the residue theorem, but that seems to be a lot of work so if anyone has a better idea I'd be glad to hear it.
78517
Inspired by the user @Integrals, I thought I'd find some nice integrals! Especially interesting are those involving $\log \pi$. From Borwein and Devlin's "The Computer as Crucible", pg. 58 - show that  $$\displaystyle \int_0^{\infty} \frac{\log x}{\cosh^2x} \ \mathrm{d}x = \log\frac {\pi}4 - \gamma,$$  where $\gamma$ is the Euler-Mascheroni constant.
68004
$$\log \Gamma (n)=n\log n -n +\frac{1}{2} \log \frac{2\pi}{n}+\int_0^\infty \frac{2\arctan (\frac{x}{n})}{e^{2\pi x}-1} \,\mathrm{d}x$$  Is an identity that is derived from using Sterling's approximation. I can't quite figure out how it was used, and was wondering for a proof.
48568
Evaluate $$\int_0^{+\infty}\frac{\log x}{1+e^x}\,dx.$$  I have tried using Feynman's Trick (in several ways, but for example by introducing a variable $a$ such that $I(a)=\int_0^{+\infty}\frac{\log ax}{1+e^x}\,dx$), but that doesn't seem to work. Also integration by parts and all kinds of substitutions make things worse (I have no idea how to substitute such that $\log$ and $\exp$ both become simpler.  (Source:  - Level 5/5)
120960
Prove $$\int_0^\infty \frac{(\pi x - 2\log{x})^3}{\left(\log^2{x} + \frac{\pi^2}{4}\right)(1+x^2)^2} \text{d}x = 8\pi$$   I tried using a modified version of the integrand and an origin-indented semicircular contour on the positive real half of the complex plane.  Despite my best efforts, I am unable to retrieve the desired integral, as the negative factor from the negative imaginary axis is preventing me from being able to simplify the integrals.  On top of that, the residue doesn't come anywhere close to the exact result.  Are there better methods that I am missing?
102100
Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution )  1)$$\int_0^\infty { e^{-x^2} \ln x }\,dx = -\tfrac14(\gamma+2 \ln 2) \sqrt{\pi} $$  2)$$\int_0^\infty { e^{-x} \ln^2 x }\,dx = \gamma^2 + \frac{\pi^2}{6} $$  3) $$\gamma = \int_0^1 \frac{1}{1+x} \sum_{n=1}^\infty x^{2^n-1} \, dx$$  and lastly,  4) $$\zeta(s) = \frac{e^{(\log(2\pi)-1-\gamma/2)s}}{2(s-1)\Gamma(1+s/2)} \prod_\rho \left(1 - \frac{s}{\rho} \right) e^{s/\rho}\!$$  I personally think the last is obtained from a simple use of the Weierstrass factorization theorem. I'm unsure as to what substitution is used.  $\gamma$ is the Euler-Mascheroni constant and $\zeta(s)$ is the Riemannian Zeta function.  Thanks in advance.
95551
I'd like some help with the following integral:  $$\int_0^\infty \lfloor x \rfloor e^{-x}\mathrm dx .$$  Thanks.
121772
What is the value of   $$\int\frac{1}{\log x}dx$$ I have tried many times, but failed everytime.   Can anyone help me out in solving this question.
37901
Does the laplace transform of $1/t$ exist? If yes, how do we calculate it? Putting it in $$\int_0^\infty (e^{-st}/t) dt$$ won't solve.  Is there any other way? If not, why?  Thanks!
12062
I have no idea where to even start. WolframAlpha cant compute it either.  $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$  I think it can be done with series, but I am not sure, can someone help a little so I can get a start??  Thanks!
71054
It always puzzles me, how the Gamma function's inventor came up with its definition $$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$ Is there a nice derivation of this generalization of the factorial?
75056
why: $$\int_0^1\int_{u(t)}^{u(t)+w(t)} f(t,v(t)) dv dt = \int_0^1 f(t,u(t)+\theta w(t))w(t) dt; ~~\theta\in[0,1] $$  how to get this ?  Please help me   Thank you.
116368
I don't know how to evaluate it. I know there is one method using the gamma function. BUT I want to know the solution using a calculus method like polar coordinates.  $$\int_{-\infty}^\infty x^2 e^{-x^2}\mathrm dx$$  I will wait for a solution. Thank you.
62560
$$ \sum_{n=1}^\infty \frac{n!e^n}{n^n} $$ Where $e$ is Euler's number.  Recently on calculus class we were covering convergence tests and my group got stuck with this infinite series. Our calculus teacher told us that it diverges but he doesn't see any elementary way or simple test to prove it. He told us something about Euler's Gamma function and said that he knows it diverges because of Stirling's formula but this is not quite our level yet.  I tried to think about it myself but I established only that this  is some kind of an edge case. We could generalize the problem to series of $\frac{n!a^n}{n^n}$. Then, from the ratio test we know that it converges for $a &lt;e$ and diverges for $a&gt;e$. For $a=e$ the test is inconclusive. I've also tried to compare it with various divergent series, but nothing worked...  Does anybody have any idea for a simple solution?
56394
I do not know how to solve this integral and then expand the result in powers of $t$  $$(1-t)^{-1/2}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{2x^2t}{1+t}\right)}{1+x^2}dx=?,~~~~|t|&lt;1.$$  any idea?
110711
$$\int_{-\infty}^{\infty}{{\rm d}x \over 1 + x^{2n}}$$  How to calculate this integral? I guess I need to use residue. But I looked at its solution. But it seems too complicated to me. Thus, I asked here. Thank you for help.