How might I evaluate this integral: $$\int_0^\infty \frac{\log\ x}{1+x^2}dx$$  I tried substituing $x = \tan\ v$ but that didn't get me anywhere. Any suggestions as to a thought process for this one?
123571
I want to show that $\int_{0}^{\infty}\frac{\log x}{x^2+1}dx=0$, but is there a faster method than finding the contour and doing all computations?  Otherwise my idea is to do the substitution $x=e^t$, integral than changes to $\int _{-\infty}^{\infty}\frac{t e^t}{1+e^{2t}}dt$. Next step is to take the contour $-r,r,r+i\pi,-r+i\pi$ and integrate over it...
118451
My question is, how can I evaluate the following integral?  $$\int\frac1{\sqrt[4]{1+x^4}}\mathrm dx$$  Thanks.
88604
Essentially, all is in the title: Is it possible to compute the integral $$\int_1^{+ \infty} \frac{dx}{x^2(1+e^x)} \hspace{1cm} ?$$  I suspected some relation with a , but I was not able to find something explicit.
109728
Please help me to find a closed form for the following integral: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$  I was told it could be calculated in a closed form.
1209
Evaluate the integral $$\int_0^{\pi}\frac{a^n\sin^2x+b^n\cos^2x}{a^{2n}\sin^2x+b^{2n}\cos^2x}dx.$$   I have no idea.   $$\int_0^{\pi}\dfrac{a^n\sin^2x+b^n\cos^2x}{a^{2n}\sin^2x+b^{2n}\cos^2x}dx=\int_0^{\pi/2}\dfrac{a^n\sin^2x+b^n\cos^2x}{a^{2n}\sin^2x+b^{2n}\cos^2x}dx=\int_0^{\pi/2}\dfrac{a^n\tan^2x+b^n}{a^{2n}\tan^2x+b^{2n}}dx$$  I try the substitution $\tan{x}=t$, but it doesn’t work.
79865
I need a little help with this question, please!   I have to evaluate the real convergent improper integrals using RESIDUE THEORY (vital that I use this), using the following contour:  $$\int\limits_0^\infty \frac{\log x} {(1+x^2)^2} dx$$    Using this contour:     $R&gt;1$ and $r&lt;1$
33350
Some time ago I came across  to the following integral:    $$I=\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$$ What are the hints on how to compute this integral?
63223
Given the integral $\int_{0}^{\infty} \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})} dx$  How can I evaluate it? Thanks!
74195
Good afternoon, I need to find:  $$\int_0^\infty\frac{x^n}{ax^2+bx+c}\space dx$$  I've no idea about any clue what so ever.
102145
i'm required to evaluate this integral. I've tried factorizing but it doesn't lead me to anywhere.   $$\int\frac{x^2-1}{(x^4+3 x^2+1) \tan^{-1}\left(\frac{x^2+1}{x}\right)}\,dx$$  I've also tried letting $u = \frac{x^2+1}{x}$, $du/dx$ gets me $1-\frac{1}{x^2}$ but it doesn't seem to be working either.  Hope to receive some advise/ solutions on how to start tackling the question
40112
Evaluate $$\int_0^\infty\frac{\ln x}{1+x^2}\ dx$$ I don't know where to start with this so either the full evaluation or any hints or pushes in the right direction would be appreciated.  Thanks.
60941
How do I integrate the following function: $$\int_{0}^{\infty} \frac{\log{x}}{x^{2}+a^{2}} \ dx $$ Integrating by parts looks very difficult.
31614
How can I evaluate $$I=\int_{0}^{+\infty}\!e^{-ax^2-\frac b{x^2}}\,dx$$ for $a,b&gt;0$?  My methods:  Let $a,b &gt; 0$ and let $$I(b)=\int_{0}^{+\infty}e^{-ax^2-\frac b{x^2}}\,dx.$$ Then $$I'(b)=\int_{0}^{\infty}-\frac{1}{x^2}e^{-ax^2-\frac b{x^2}}\,dx.$$  What the other methods that can I use to evaluate it? Thank you.
36126
How can I calculate the integral $I=\int_0^1\frac{\ln(1+x)}{1+x^2}\,dx$ by substituting $t=\frac{1-x}{1+x}$.
116065
How would one evaluate the integral $$\int_0^\infty \frac{\arctan(x) \,dx}{x(1+x^2)}$$?  I was told it had a nice closed form and could have been solved with differentiation under the integral sign; however, I tried to set $$I(\alpha) = \int_0^\infty \frac{\arctan(\alpha x) \,dx}{x(1+x^2)}$$ and got nowhere (the resulting integral was very messy). Is there a much more clever substitution that can be used to tackle the integral?
34590
How would I go about evaluating this integral? $$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$ What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex plane, excluding the negative real axis, but had trouble calculating the residue at $z=i$ (perhaps there is a way of doing this that I don't know of). After that didn't work, I tried a rectangular box integral from $\epsilon$ to $R$, from $R$ to $R+i/2$, from $R+i/2$ to $-S+i/2$, from $-S+i/2$ to $-S+i\epsilon$, from $-S+i\epsilon$ to $-\epsilon+i\epsilon$ and finally a semicircle around the origin, radius $\epsilon$.   Any help would be appreciated.
39284
Are there any ways to evaluate $\int^\infty_0\frac{\sin x}{x}dx$ without using double integral?   I can't find any this kind of solution. Can anyone please help me? Thank you.