I found this integral   And the person says this equals $ \frac{\pi ^2}{12}$  But I tried to solve this and could not do it and I was unable to find any thing online.   I tried many ways but could not even go through variable transformation by assuming ${(x^x)}^{{(x^x)}^{.^{.^.}}} = t$  How should I proceed?
96658
How can it be shown that   $$\lim_{p\to\infty}I(p)= \lim_{p \to \infty}\int^{1}_0 (x^x)^{\scriptscriptstyle {(x^x)^{(x^x)^{(x^x)^{(x^x)^{(x^x)...(p \; times)}}}}}} dx= \frac{\pi^2}{12}$$      $I(1)=\int^{1}_0 x^x dx=\sum_{n=0} ^{\infty} \frac{1}{n!}\int^{1}_0 \ln(x)^nx^n dx$   also $\int^{1}_0 \ln(x)^nx^n dx=(-1)^n (1+n)^{-1-n}n!$ .  I don't know how to calculate $I(p)$ beyond $p=1.$  Note: This integral was proposed on Romanian Mathematical Magazine and two solutions can be found .
47738
While solving another problem I have come across this integral which I am unable to evaluate. Can someone please evaluate the following integral? Thank you.      $$\int_0^{2\pi}\frac{1}{(2+\cos\theta)^2}\,d\theta.$$
111619
I need help evaluating the following integral $$\int_{0}^{\infty} \frac{\sin^4 x}{x^2}\,dx$$ which should probably be equal to $\frac{\pi}{4}$  Using some trigonometric manipulations I got $\frac{3}{8} - \frac{\cos{2x}}{2} + \frac{\cos{4x}}{8}$ which using integration by parts doesn't lead me to anything pretty.  Update: Not sure if I should post this as a separate question but getting explanation why $\int_{0}^{\infty} \frac{\sin{ax}}{x} = \frac{\pi}{2}$ for a positive integer $a$ could help me solve this question.
85817
How do I solve this? $$\int_0^\infty\frac{\ln(1+{x^2})}{1+{x^2}}$$ I know the answer is $\pi\ln2$.
65916
I stumbled upon this integral $$I=\int_1^3 \frac{\log x}{x^2+3}dx$$ Since the answer given by Wolfram-Alpha is $I=\frac{\pi \log 3}{12\sqrt3}$ I tried to work near the bounds of the integral because as a indefinite integral it will involve some polylogarythms which I want to avoid. But not a single substitution  like $x=\frac{1}{y}$ or $x-2=u$ gave any better look to this integral. And when integrated by parts I feel like it got worse. Could you perhaps help me on this?
89872
How do you go about integrating this? For context purposes, $s$ is a constant and $n$ is just the variable we need to integrate for. If needed, a number can be substituted for $s$, but I would like it without preferably.   $$\int\frac{s\sqrt{3}\sqrt{s^2-2ns\sqrt{7}+4n^2}}{2s^2-4ns\sqrt{7}+8n^2}dn$$
56366
I was reading an article in which it was stated that, with a change of variable, one could show that:  $$\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$$  I tried with $t = 1 + \frac{1}{x}$ but that doesn't work out, especially because the lower bound doesn't become $0$.
106957
$$\int_0^\pi x \ln(\sin (x))dx  $$   I tried integrating this by parts but I end up getting integral that doesn't converge, which is this $$ \int_0^\pi \dfrac{x^2\cos (x)}{\sin(x)} \ dx$$ So can anyone help me on this one?
86618
I am unable to solve the following integral: $$\int \frac{\sec^{2}{x}}{(\sec{x} + \tan{x})^{5/2}} \ \mathrm dx.$$   Tried putting $t=\tan{x}$ so that numerator has $\sec^{2}{x}$ but that doesn't help.
28927
Integrate $$\int \frac{1}{1+\sqrt{x^2+2x+2}}dx$$   I have tried by using Euler substitution, but that gave me a wrong answer.  So can somebody help?
20744
I am in the middle of a problem and having trouble integrating the following integral:  $$\int_{-1}^1\frac1{(1+x^2)^2}\mathrm dx$$  I tried doing partial fractions and got:  $$1=A(1+x^2)+B(1+x^2)$$  I have no clue how to solve this since it is obvious there is no way to cancel out either $A$ or $B$ to get the other variable. Please guide me.  Thank you.
103492
Computing the integration.  $\int_0^1\frac{\ln x}{1-x}d x$  I have no idea ... how to do that?  ... The answer is $-\frac{\pi ^2}{6}$ I get it by software.
25858
I want to know how to integrate  $$\int\frac{\sqrt{1+x^2}}{x}\,\mathrm dx$$  Could anyone solve it?  Thanks
108659
How do I compute the integration for $a&gt;0$,   $$ \int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx? $$  I want to find a complex function and integrate by the .
39298
I'd love your help with finding the sign of the following integral: $\int_{0}^{2 \pi}\frac{\sin x}{x} dx$,I know that computing it is impossible.  I tried to use integration by parts and maybe to learn about the sign of each part and conclude something but It didn't work for me.  Any suggestions?
941
$$\int\sqrt{x^2+1}dx$$ I've been attempting this problem for days and I've made very little progress. I understand that I should substitute $x=\tan(x)$ and $dx=\sec^2(x)dx$ but beyond that I am very much lost.
11731
How you integrate $\frac{1}{\sqrt{1+x^2}}$ using following substitution? $1+x^2=t$ $\Rightarrow$ $x=\sqrt{t-1} \Rightarrow dx = \frac{dt}{2\sqrt{t-1}}dt$... Now I'm stuck. I don't know how to proceed using substitution rule.