Question

What is the value of \(\int_0^{π/2}\sqrt{sin⁡θdθ} + \int_0^{π/2}\sqrt{cos⁡θ⁡dθ} \)?

(a) \(8\sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} \)

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

(c) \(8\sqrt{\pi} \frac{\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})} \)

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

The question was asked during an online exam.

I would like to ask this question from Special Functions topic in chapter Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

Correct choice is (a) \(8\sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} \)

To explain: \(\int_0^{π/2}\sqrt{sin⁡θdθ} + \int_0^{π/2}\sqrt{cos⁡θ⁡dθ} \)

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

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

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

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

\( = 8\sqrt{\pi} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}. \)