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Questions  >>  CBSE XII  >>  Math  >>  Integrals
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Q)

$\begin{align*}\int_0^\frac{\pi}{2} \frac{\sin x}{1 + \cos^2 x } dx \end{align*}$


$(A)\; \frac{\pi}{2}$
$(B)\; \frac{\pi}{4}$
$(C)\; \frac{\pi}{6}$
$(D)\;\frac{\pi}{3}$

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A)
Put $\cos x = t$
$ \implies - \sin x\; dx = dt$
$\implies \sin x\; dx = -dt$
$\therefore \begin{align*} I = \int_1^0 \frac{dt}{1+t^2} \end{align*} $
$\begin{align*} I = \int_0^1 \frac{dt}{1+t^2}\end{align*}$
$ =[\tan^{-1} t]_0^1$
$= \tan^{-1} (0) - \tan^{-1}(0)$
$=\frac{\pi}{4}$
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