## Some interesting integrals I saw online

Posted: August 2, 2012 in Mathematics

(1) Evaluate $\displaystyle\int\sin(101x)\sin^{99}x\;dx$

(2) Evaluate $\displaystyle\int_0^1\frac{\arctan{x}}{x+1}\;dx$

(3) For $n>1$, prove that $\displaystyle\int_0^\infty\frac{dx}{(x+\sqrt{1+x^2})^n}=\frac{n}{n^2-1}$

(4) If $f$ is a bounded non-negative function, then show that
$\displaystyle\int_0^\infty f\left(x+\frac{1}{x}\right)\frac{\log{x}}{x}dx=0$

(5) Evaluate $\displaystyle\int_0^1\log(\sqrt{1-x}+\sqrt{1+x})\;dx$

(6) Evaluate $\displaystyle\int_{-\pi/2}^{\pi/2}\frac{1}{2007^x+1}\cdot\frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}\;dx$