Mark the correct alternative in each of the following:
Question:

Mark the correct alternative in each of the following:

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

A. $\frac{1}{5 \mathrm{x}}\left(4+\frac{1}{\mathrm{x}^{2}}\right)^{-5}+\mathrm{C}$

B. $\frac{1}{5}\left(4+\frac{1}{x^{2}}\right)^{-5}+C$

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

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

Solution:

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

$I=\int \frac{x^{9}}{x^{12}\left(4+\frac{1}{x^{2}} 6\right.} d x$

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

Let $\left(4+\frac{1}{x^{2}}\right)=t ; \frac{-2}{x^{2}} d x=d t$

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

$I=\frac{1}{10}\left[\frac{1}{t^{5}}\right]$

$I=\frac{1}{10}\left(\left[4+\frac{1}{x^{2}}\right]^{-5}\right)+c$

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