Prove the following and use it to evaluate the integral:
8829
How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$
68821
Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$
67533
A question on integration
115598
Evaluating $\int_{0}^{1} dx\frac{\log(1+x)}{1 + x^2}$
8358
Prove $\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $
131120
Proving an integral equation.
63167
Evaluate integral $\int_0^\frac{\pi}{2} \ln\left(\frac{1+a\cos x}{1-a\cos x}\right) \frac{\mathrm{d} x}{\cos x}$ for $\left|a\right|<1$
123695
Infinite series: Integral test( Proof )
30459
Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$
84765
Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$
24808
How to evaluate the integral $\int \sqrt{1+\sin(x)} dx$
116778
Calculation of an integral via residue.
73240
How do I prove this form of mean value theorem for integral?
25509
evaluation of an integral involving algebraic numbers
89028
Evaluate the integral $\int e^{-x}x^n dx$
156531
Evaluating the integral $\int_{-\infty}^\infty \frac{\sin^2(x)}{x^2}e^{i t x} dx$
1588
How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$