Question

What is the value of \(\int_0^∞ (sech x)^5  dx \)?

(a) \(\frac{3\pi}{80} \)

(b) \(\frac{3\pi}{240} \)

(c) \(\frac{3\pi}{16} \)

(d) \(\frac{\pi}{240} \)

This question was posed to me in a job interview.

I'm obligated to ask this question of Special Functions topic in division Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

Correct option is (c) \(\frac{3\pi}{16} \)

The best I can explain: \(\int_0^∞ (sech x)^5  dx = \frac{2^5}{4} \beta(\frac{5}{2}, \frac{5}{2}) \)

\(\displaystyle = \frac{8 \Gamma(\frac{5}{2})\Gamma(\frac{5}{2})}{\Gamma(5)} = \frac{(8*\frac{3}{2}*\frac{1}{2}*\sqrt{\pi}*\frac{3}{2}*\frac{1}{2}*\sqrt{\pi})}{24} = \frac{9π}{48}=\frac{3π}{16}. \)