Proving the asymptotic of $\int_0^\infty\frac{\sin x}{x^{a+1}}\,dx$ for $a \to 0^+$ is $\pi/2$
67953
Finding the asymptotic of an integral
53307
how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$
89602
$ \int_{0}^{ \infty} \int_{0}^{ \infty} \frac { e^{-(x+y)}}{x+y} dx dy $
30193
Real-Analysis Methods to Evaluate $\int_0^\infty \frac{x^a}{1+x^2}\,dx$, $|a|<1$.
28771
Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $
117264
Prove that $\int_0^{\infty} \frac{\sin x}{x^p}\, dx$ converges for $0<p<2$
141892
How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$
122843
Finding $\int_0^\infty \frac{\cos(ax)-\cos(bx)}{x}dx$
138169
Finding $\lim\limits_{n \to \infty }n\int_{0}^{1}\frac{x^{n}}{1+x+x^{n}}dx$
147608
Cauchy principle value of $\int_{-\infty}^{\infty}\sin(x)/(x-a)dx$
68097
Prove $\int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{2}\sqrt{\frac{\pi}{a}}\mathrm{e}^{-2\sqrt{ab}}$
65407
Evaluating $\int\limits_0^\infty \frac{\log x} {(1+x^2)^2} dx$ with residue theory
129892
show $\int_0^{\infty} \frac{x\sin ax}{x^2+t^2}dx = \frac{\pi}{2}e^{-at}$
5537
Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$
27002
Evaluating $\int\limits_0^{\pi} \frac{dx}{1+2\sin^2x}$
148743
$\int_0^\infty \frac{1}{1+x^ 9} \, dx$
69978
Evaluating $\int_1^\infty \frac{1}{x(x^2+1)}\ dx$