First slide

Evaluation of definite integrals

Question

Let $I_{n} = \int_{0}^{π / 2} x^{n} \cos x d x$ then $I_{8} + 56 I_{6}$ is equal to

Easy

A

${(\frac{π}{2})}^{6}$
B

${(\frac{π}{2})}^{8}$
C

${(\frac{π}{2})}^{6} - 1$
D

${(\frac{π}{2})}^{8} - 1$

Solution

$\begin{array}{l} I_{n} = \int_{0}^{π / 2} x^{n} \cos x d x \\ = {x^{n} \sin x|}_{0}^{π / 2} - n \int_{0}^{π / 2} x^{n - 1} \sin x d x \\ = {(\frac{π}{2})}^{n} - n [- {x^{n - 1} \cos x|}_{0}^{π / 2} + (n - 1) \int_{0}^{π / 2} x^{n - 2} \cos x d x] \\ = {(\frac{π}{2})}^{n} - n (n - 1) I_{n - 2} \\ I_{8} + 8.7 I_{6} = {(\frac{π}{2})}^{8} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App