Evaluate : 1 8+3 x-x^2 d x. - Vidyadip

Question

Evaluate : $\int \frac{1}{\sqrt{8+3 x-x^2}} d x$.

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Answer

$\begin{aligned} I & =\int \frac{d x}{\sqrt{8+3 x-x^2}} \\ & =\int \frac{d x}{\sqrt{-\left(x^2-3 x-8\right)}} \\ & =\int \frac{d x}{\sqrt{-\left(x^2-2\left(\frac{3 x}{2}\right)+\frac{9}{4}-\frac{9}{4}-8\right)}} \\ & =\int \frac{d x}{\sqrt{-\left(\left(x-\frac{3}{2}\right)^2-\frac{41}{4}\right)}} \\ & =\int \frac{d x}{\sqrt{\left(\frac{\sqrt{41}}{2}\right)^2-\left(x-\frac{3}{2}\right)^2}}\end{aligned}$
$\begin{aligned} & =\sin ^{-1}\left(\frac{x-\frac{3}{2}}{\frac{\sqrt{41}}{2}}\right)+c \\ \therefore I & =\sin ^{-1}\left(\frac{2 x-3}{\sqrt{41}}\right)+c\end{aligned}$

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