Question

What is the value of \(\int_0^∞ \frac{1}{(1+x^4 )} dx\)?

(a) \(\frac{\sqrt{2} \pi}{4} \)

(b) \(\frac{\sqrt{3} \pi}{6} \)

(c) \(\frac{\sqrt{2} \pi}{6} \)

(d) \(\frac{\sqrt{3} \pi}{4} \)

I got this question by my school principal while I was bunking the class.

Enquiry is from Special Functions topic in section Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

Correct answer is (a) \(\frac{\sqrt{2} \pi}{4} \)

Easy explanation: Substitute \(x^2= tan(θ) \)

\(2xdx = (sec(θ))^2dθ \)

= \(\frac{1}{2} * \int_0^∞ \frac{1}{\sqrt{sin(θ)cos(θ)}} dθ \)

= \(\frac{1}{4} * \beta(\frac{1}{4}, \frac{3}{4}) \)

= \(\frac{1}{4} * \frac{\Gamma(1⁄4).\Gamma(3⁄4)}{\Gamma(1)} \)

= \(\frac{1}{4} * \frac{\pi}{sin(π/4)} \)

= \(\frac{\sqrt{2} \pi}{4}.\)