First slide

Methods of integration

Question

$\int \frac{x^{5 m - 1} + 2 x^{4 m - 1}}{{(x^{2 m} + x^{m} + 1)}^{3}} d x = f (x) + c$ then $f (x) =$

Moderate

A

$\frac{x^{5 m}}{2 m {(x^{2 m} + x^{m} + 1)}^{2}}$
B

$\frac{x^{4 m}}{2 m {(x^{2 m} + x^{m} + 1)}^{2}}$
C

$\frac{2 m (x^{5 m})}{x^{2 m} + x^{m} + 1}$
D

$\frac{2 m (x^{4 m})}{x^{2 m} + x^{m} + 1}$

Solution

$\int \frac{x^{5 m - 1} + 2 x^{4 m - 1}}{x^{6 m} {(1 + x^{- m} + x^{- 2 m})}^{3}} = \int \frac{x^{- m - 1} + 2 x^{- 2 m - 1}}{{(1 + x^{- m} + x^{- 2 m})}^{3}} d x$
$\begin{matrix} p u t t = 1 + x^{- m} + x^{- 2 m} \\ \frac{d t}{d x} = - m x^{- m - 1} - 2 m x^{- 2 m - 1} \\ - \frac{d t}{m} = (x^{- m - 1} + 2 x^{- 2 m - 1}) d x \end{matrix}$

$∴ \int \frac{x^{5 m - 1} + 2 x^{4 m - 1}}{{(x^{2 m} + x^{m} + 1)}^{3}} = - \frac{1}{m} \int t^{- 3} d t = \frac{1}{2 m t^{2}} + c$

$= \frac{1}{2 m {(1 + x^{- m} + x^{- 2 m})}^{2}} + c$

$= \frac{x^{4 m}}{2 m {(x^{2 m} + x^{m} + 1)}^{2}} + c$

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