Show $\int_{0}^\infty \frac{x^q}{x^p + 1} dx = \frac{\pi}{p}\csc \frac{\pi(q+1)}{p}$, where $p>q+1$.
5537
Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$
70845
Contour method to show that $\int_0^\infty\frac{x-\sin x}{x^3} \, dx=\frac\pi4$
35319
I'm looking for several ways to prove that $\int_{0}^{\infty }\sin(x)x^mdx=\cos(\frac{\pi m}{2})\Gamma (m+1)$
4971
$\int_{0}^{1}\frac{1-x}{1+x}.\frac{dx}{\sqrt{x+x^2+x^3}}$
51837
Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
13747
Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$
74955
Simplify result of $\int_0^{\infty} \frac{1}{1+x^n}dx$
126046
solving the integral $ \int_0^\infty \dfrac{\sin xt}{x(x^2+1)} dx $
68821
Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$
37015
Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$
117264
Prove that $\int_0^{\infty} \frac{\sin x}{x^p}\, dx$ converges for $0<p<2$
27705
Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
21791
Show that $\sum\limits_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)}=\frac{1}{p!p} $ for every positive integer $p$
38515
Integral of $\frac{1}{x^4+1}$
93759
Evaluate $\int_{0}^{\infty} \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})} dx$
24716
$\int_0^\infty\frac{\log x dx}{x^2-1}$ with a hint.
34907
Show that $\int_{0}^{1}\{P(x)\}^{2}\,dx = (n + 1)^{2}\left(\int_{0}^{1}P(x)\,dx\right)^{2}$