x 9 ( 4 x 2 + 1 ) 6 ? d x is equal to:

Question

$\int \frac{x^{9}}{{(4 x^{2} + 1)}^{6}} d x$ is equal to:

Smart Achievers · Accepted Answer

(4)

Ans. $\frac{1}{10} {(4 + \frac{1}{x^{2}})}^{- 5} + C$

Explanation:

Let $I = \int \frac{x^{9}}{{(4 x^{2} + 1)}^{6}} d x$

$= \int \frac{x^{9}}{x^{12} {(4 + \frac{1}{x^{2}})}^{6}} d x$

$= \int \frac{d x}{x^{3} {(4 + \frac{1}{x^{2}})}^{6}}$

Put $4 + \frac{1}{x^{2}} = t$

$\Rightarrow$ $\frac{- 2}{x^{3}} d x = d t$

$\Rightarrow$ $\frac{1}{x^{3}} d x = - \frac{1}{2} d t$

$∴$ $I = - \frac{1}{2} \int \frac{d t}{t^{6}} = \frac{- 1}{2} \int t^{- 6} d t$

$= - \frac{1}{2} [\frac{t^{- 6 + 1}}{- 6 + 1}] + C$

$= \frac{1}{10} [\frac{1}{t^{5}}] + C$ $= \frac{1}{10} {(4 + \frac{1}{x^{2}})}^{- 5} + C$

Want Us to Email you About Special Offers & Updates?