Question

What is the value of \(\int_0^1 \frac{x^2}{\sqrt{(1-x^4 )}} \)?

(a) \(\frac{2\sqrt{\pi} \Gamma(\frac{5}{4})}{Γ(\frac{1}{4})} \)

(b) \(\frac{2\pi\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} \)

(c) \(\frac{2\sqrt{\pi} \Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} \)

(d) \(\frac{2\sqrt{\pi} \Gamma(\frac{3}{4})}{\Gamma(\frac{5}{4})} \)

I have been asked this question in an interview for job.

My question is based upon Special Functions in section Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

Correct option is (c) \(\frac{2\sqrt{\pi} \Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} \)

Best explanation: Substitute \(x^2 = sin(θ)\)

\(2xdx = cos(θ)dθ\)

= \(\int_0^{π⁄2}\frac{sin(θ)}{cos(θ)}   \frac{cos(θ)}{2\sqrt{sin(θ)}}  dθ \)

= \(\frac{1}{2} * \beta( \frac{3}{4} , \frac{1}{2}) \)

= \(\frac{\frac{1}{2} *  \Gamma(\frac{3}{4}) *  \Gamma(\frac{1}{2}) }{\Gamma(\frac{5}{4})} \)

= \(\frac{2\sqrt{\pi} \Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} \).