Archive for August, 2012

(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

Advertisements