Integrate the function: x 9 - 4 x^2 - Vidyadip

Question

Integrate the function: $ \frac{x}{{9 - 4{x^2}}}$

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Answer

Let $I = \int {\frac{x}{{9 - 4{x^2}}}} dx$$ = \frac{{ - 1}}{8}\int {\frac{{ - 8x}}{{9 - 4{x^2}}}dx} $ ...(i)
Putting $9 - 4x^2 = t$
$ \Rightarrow - 8x = \frac{{dt}}{{dx}}$
$\Rightarrow - 8xdx = dt$
$\therefore$ From eq. (i), $I = \frac{{ - 1}}{8}\int {\frac{{dt}}{t} }$
$ = \frac{{ - 1}}{8}\log \left| t \right| + c$
$= \frac{{ - 1}}{8}\log \left| {9 - 4{x^2}} \right| + c$

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