Question

Find the value of \(\int_{0}^{\frac{\pi}{2}} cos^{11}(x).sin^9(x)\, dx\).

(a) ^1⁄10!

(b) ^5!6!⁄11!

(c) ^10!⁄5!6!

(d) 0

This question was posed to me in an internship interview.

My question is based upon Integral Reduction Formula topic in portion Integral Calculus of Engineering Mathematics

Answer 1

The correct option is (b) ^5!6!⁄11!

The explanation is: Using the definition of beta function we see that the integral is equal to the beta function at (6,5)

Now using the relation between the Beta and the Gamma function we have

\(\beta(m, n)=\frac{\Gamma (m).\Gamma (n)}{\Gamma (m+n)}\)

\(\beta(6, 5)=\frac{\Gamma (6).\Gamma (5)}{\Gamma (11)}=\frac{6!.5!}{11!}\)