Question

What is the value of the integral \(\int_0^{π⁄2}\sqrt{tan(θ) } \, dθ\)?

(a) \(\frac{\Gamma(\frac{3}{4})^2}{\sqrt{\pi}} \)

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

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

(d) \(\frac{\Gamma(\frac{1}{4})^2}{\pi} \)

I have been asked this question in class test.

Origin of the question is Special Functions in section Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

Correct choice is (a) \(\frac{\Gamma(\frac{3}{4})^2}{\sqrt{\pi}} \)

Explanation: \(\int_0^{π⁄2}\sqrt{tan(θ)} \, dθ \)

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

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

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

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