d x 9 x-4 x^ 2 equals - Vidyadip

Question

$\int \frac{d x}{\sqrt{9 x-4 x^{2}}}$ equals

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Answer

$\int \frac{d x}{\sqrt{9 x-4 x^{2}}}=\int \frac{d x}{\sqrt{-4\left(x^{2}-\frac{9}{4} x\right)}}$
$=\int \frac{d x}{\sqrt{-4\left(x^{2}-\frac{9}{4} x+\frac{81}{64}-\frac{81}{64}\right)}}$
$=\int \frac{d x}{\sqrt{-4\left[\left(x-\frac{9}{8}\right)^{2}-\left(\frac{9}{8}\right)^{2}\right.}]}$
$=\frac{1}{2}\left[\sin ^{-1}\left(\frac{x-\frac{9}{8}}{\frac{9}{8}}\right)\right]+C$
$=\frac{1}{2}\left[\sin ^{-1}\left(\frac{8 x-9}{9}\right)\right]+C$

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