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

Question

Evaluate: $\int_0^{\frac{\pi}{2}} \frac{\sin 2 x}{1+\sin ^2 x} d x$

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Answer

$\left.\int_0^{\frac{\pi}{2}} \frac{\sin 2 x}{1+\sin ^2 x} d x=\left[\log \left|1+\sin ^2 x\right|\right]_0^{\frac{\pi}{2}} \quad \ldots \ldots . . \cdot \frac{ f ^{\prime}(x)}{ f (x)} d x=\log | f (x)|+ c \right]$
$=\log \left|1+\sin ^2\left(\frac{\pi}{2}\right)\right|-\log \left|1+\sin ^2 0\right|$
$=\log |1+1|-\log 1$
$=\log 2-0$
$=\log 2$

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