Question

Find \(\int_{0}^{\frac{\pi}{4}} x^4.sin(x)dx\).

(a) -1

(b) 1

(c) 0

(d) 4((^π⁄2)^3 – 3π + 1)

This question was posed to me at a job interview.

This interesting question is from Integral Reduction Formula in chapter Integral Calculus of Engineering Mathematics

Answer 1

Right option is (b) 1

Explanation: Using the formula

\(\int_{0}^{\frac{\pi}{2}} x^a.sin(x)dx=\left ((\frac{\pi}{2})^{a-1}-(a-1)\int x^{a-2}sin(x)\right)\)

We have

\(\int_{0}^{\frac{\pi}{2}} x^4.sin(x)dx=4.\left ((\frac{\pi}{2})^{3}-3\int_0^{\frac{\pi}{2}} x^{2}sin(x)dx\right)\)

4((π/2)^3-3π+1)