2008-31.md

course	course_year	question_number	tags	title	year
Complex Analysis or Complex Methods	IB	31	IB 2008 Complex Analysis or Complex Methods	1.II.13C	2008

By a suitable choice of contour show the following:

(a)

$$\int_{0}^{\infty} \frac{x^{1 / n}}{1+x^{2}} d x=\frac{\pi}{2 \cos (\pi / 2 n)}$$

where $n>1$,

(b)

$$\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x=\frac{\pi^{2}}{2 \sqrt{2}}$$