Prove $\int_{0}^{\infty} \frac{x^{p-1}}{1+x} dx = \frac{\pi}{\sin (p \pi)}$ without using residue theorem or Beta function
30193
Real-Analysis Methods to Evaluate $\int_0^\infty \frac{x^a}{1+x^2}\,dx$, $|a|<1$.
43368
How to prove $\sum\limits_{n=1}^\infty\frac{\sin(n)}n=\frac{\pi-1}2$ using only real numbers.
49074
Compute $\int\limits_{-\infty}^{\infty}\frac{1}{(1+x^2)^{n+1}}dx$ via residue calculus.
109606
Detail in the use of the Residue Theorem for $\int\limits_{-\infty}^{\infty}\frac{\cos(2x)-1}{x^2}dx$
16775
Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$
53307
how to strictly prove $\sin x<x$ for $0<x<\frac{\pi}{2}$
68421
Residue theorem with $\frac{1}{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) d\theta$.
29546
Residue integral: $\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx$ with $0 \lt a \lt 1$.
21375
The integral $\int_0^\infty \dfrac{x \sin(x)}{x^2+1} dx$
8829
How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$
101587
Calculating $\int_{- \infty}^{\infty} \frac{\sin x \,dx}{x+i} $
51284
$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus
50509
Prove that $\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}}~dx=0$ (without trigonometric substitution)
58646
Are there any ways to evaluate $\int^\infty_0\frac{\sin x}{x}dx$ without using double integral?
129892
show $\int_0^{\infty} \frac{x\sin ax}{x^2+t^2}dx = \frac{\pi}{2}e^{-at}$
62109
$\int_{-\infty}^{\infty} \frac{\cos(αx)}{(x^2+1)(x^2+4)} \mathrm dx$ using Complex methods
83703
Show that $\int_{0}^{\infty}|\frac{\sin x}{x}|\,dx=\infty $