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

$\begin{align*}\int_0^1 \frac{\tan^{-1} x}{1+x^2} dx \end{align*}$


$(A)\; \frac{\pi}{32}$
$(B)\; -\frac{\pi^2}{32}$
$(C)\; -\frac{\pi}{32}$
$(D)\; \frac{\pi^2}{32}$

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A)
Let $\tan^{-1} x = t \;\;\;\;\;\;when x = 0 , t = 0$
$\; \; \; \frac{1}{1+x^2} dx = dt \; \; \; \; \; \; \; \; \; x = 1, t = \frac{\pi}{4}$
$I = \int_\frac{\pi}{2}^0 t\; dt = [\frac{t^2}{2}]_0^\frac{\pi}{2}$
$ \; \; = \frac{1}{2} [\frac{\pi^2}{16} - 0] = \frac{\pi^2}{32} $
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