Question

Find \(\int_{0}^{\frac{\pi}{2}} sin^{10}(x)cos(x)dx\).

(a) 1

(b) 0

(c) \(\frac{13\pi}{1098}\)

(d) \(\frac{21\pi}{2048}\)

I have been asked this question during a job interview.

This intriguing question originated from Integral Reduction Formula topic in chapter Integral Calculus of Engineering Mathematics

Answer 1

Correct answer is (d) \(\frac{21\pi}{2048}\)

Easiest explanation: Rewriting the function as

=\(\int_{0}^{\frac{\pi}{2}} (sin^{10}(x)(1-sin^2(x)))dx=\int_{0}^{\frac{\pi}{2}} sin^{10}(x)dx-\int_{0}^{\frac{\pi}{2}} sin^{12}(x)dx\)

We now apply the formula seperately for the two integrals

\(\frac{9.7.5.3.\pi}{10.8.6.4.2.2}-\frac{11.9.7.5.3.\pi}{12.10.8.6.4.2.2}\)

\(\frac{9.7.5.3.\pi}{10.8.6.4.2.2}\times (1-\frac{11}{12})=\frac{21\pi}{2048}\)