Evaluate: _0^1 1 3+2 x-x^2 d x - Vidyadip

Question

Evaluate: $\int_0^1 \frac{1}{\sqrt{3+2 x-x^2}} d x$

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Answer

$\text { Let } I =\int_0^1 \frac{1}{\sqrt{3+2 x-x^2}} d x$
$=\int_0^1 \frac{1}{\sqrt{4-1+2 x-x^2}} d x$
$=\int_0^1 \frac{1}{\sqrt{4-\left(x^2-2 x+1\right)}} d x$
$=\int_0^1 \frac{1}{\sqrt{(2)^2-(x-1)^2}} d x$
$\left.=\sin ^{-1}\left(\frac{x-1}{2}\right)\right]_0^1(0)-\sin ^{-1}\left(\frac{1}{2}\right)$
$=0-\left(-\frac{\pi}{6}\right)$
$\therefore I =\frac{\pi}{6}$

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