I want to prove that:$$\int_{-\infty}^\infty f(x)dx=\int_{-\infty}^\infty f\left(x-\frac1x\right)dx$$ And use the result of this proof to evaluate:$$\int_{-\infty}^\infty\frac{x^2}{x^4+1}dx$$
69480
If $f(x)$ is a continuous function on $(-\infty, +\infty)$ and $\int_{-\infty}^{+\infty} f(x) \, dx$ exists. How can I prove that $$\int_{-\infty}^{+\infty} f(x) \, dx = \int_{-\infty}^{+\infty} f\left( x - \frac{1}{x} \right) \, dx\text{ ?}$$
49669
How to show that :  $I_n = \int_{-1}^{1} (1-x^2)^n dx $ is equal to $\dfrac{2^{2n+1}(n!)^2}{(2n+1)!}$ ?  With integration by parts ? I don't know how to prove this equality. Someone could help me ? Thank you in advance :)
42380
I have in trouble for evaluating following integral  $$\int_0^{\infty}  \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$$  It seems really easy, but I don't know how to handle it at all. (The results are well known, here I tried to evaluate it but I failed)  I tried to use the relation   $$\sqrt{1+x^{4}}-x^{2}=\frac{1}{\sqrt{1+x^{4}}+x^{2}}$$   but I couldn't find the desired results.
2645
How to prove  $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
3970
I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$ What do you think?  It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm dx$$ is also $\frac{\pi}{2}.$ Many proofs of this latter one are already in this .
40135
Make a substitution to express the integrand as a rational function and then evaluate the integral  $$\int {\frac{2}{x\sqrt{x+1}}}\, dx$$  What is the substitution that I have to make?
35398
I need help with this exercise.  I need to prove $$\int_{0}^{1}x^{-x}\ dx=\sum_{n=1}^{\infty}n^{-n}$$  I think I should use some convergence theorem, but I'm stuck.  Thanks a lot!
42140
How can you evaluate $\sum_{n=1}^\infty nx^{n-1} = \frac{1}{(1-x)^2}$ without relying on the fact that it's the derivative of $\sum_{n=1}^\infty x^n = \frac{1}{1-x} $?
118591
We know that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\int_0^\infty \frac{\sin x}{x} \, dx=\frac{\pi}{2}.$$ How do I show that $$\int_0^\infty \left\vert\frac{\sin x}{x} \right\vert \, dx$$ converges?
22882
How to prove that  $$\sum_{n=1}^{\infty}\frac{1}{4^n\cos^2\frac{x}{2^n}}=\frac1{\sin^2x}-\frac1{x^2}$$
107304
The problem is to show  $$\int_0^\infty \frac{e^{-x}-e^{-xt}}{x}dx = \ln(t),$$  for $t \gt 0$.  I'm pretty stuck. I thought about integration by parts and couldn't get anywhere with the integrand in its current form. I tried a substitution $u=e^{-x}$ and came to a new integral (hopefully after no mistakes)  $$ \int_0^1 \frac{u^{t-1}-1}{\log(u)}du, $$  but this doesn't seem to help either. I hope I could have a hint in the right direction... I really want to solve most of it by myself.  Thanks a lot!
35866
Evaluate $$\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \ dx$$
4428
Let $f$ be a $C^2$ function in $[0,1]$ such that $f(0)=f(1)=0$ and $f(x)\neq 0\,\forall x\in(0,1).$ Prove that  $$\int_0^1 \left|\frac{f{''}(x)}{f(x)}\right|dx\ge4$$
54941
$$ \int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2) $$  Anyone an idea on how to prove this?
34591
I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\in \mathbb{N}^+\setminus\{1\}$   I wonder if it is possible by using only real analysis to demonstrate this "innocent" result?  Edit A more general result showing that  $$\int\limits_{0}^{\infty} \frac{x^{a-1}}{1+x^{b}} \ \text{dx} = \frac{\pi}{b \sin(\pi{a}/b)}, \qquad 0 &lt; a &lt;b$$ can be found in
86529
$$\int_0^\infty x\left(\log\left(\frac1{x^2}\right)\right)^n dx$$  put $\log\left(\frac1{x^2}\right)=y^{1/n}$  $$I=\frac{\ln10}{2n}\int_{-\infty}^\infty10^{-y^{1/n}}y^{1/n}dy$$  put $10^{-y^{1/n}}=e^{-t}\implies$  $$I=\frac1{(2\ln10)^n}\int_{-\infty}^0e^{-t}t^ndt+\frac1{(2\ln10)}\int_0^\infty e^{-t}t^ndt$$  $$I=\frac1{(2\ln10)^n}\int_{-\infty}^0e^{-t}t^ndt+\frac{\Gamma(n+1)}{(2\ln10)}$$  I want to know how can I evaluate the first integral in the last step of my turn I used some substitutions so i get two integrals one of them converted to the formula of gamma function but the second could not be so because the domain of integration is different ? $n$ is a natural number
100949
Prove $$\int_0^1\prod_{n=1}^\infty(1-x^n)dx=\frac{4\pi\sqrt3}{\sqrt{23}}\frac{\sinh\frac{\pi\sqrt{23}}3}{\cosh\frac{\pi\sqrt{23}}2}.$$ In fact, product $(1-x^n)$ is difficult to compute, I hope you can show me some ideas, thank you!