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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
Question
Integrate the function w.r.t. x: $\frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}}$
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Answer

Derivative of $\tan ^{-1} x=\frac{1}{1+x^{2}}$. Thus, we use the substitution tan-1 x = t so that $\frac{d x}{1+x^{2}}=d t$ 
Therefore, $\int \frac{\sin \left(\tan ^{-1} x\right)}{1+x^{2}} d x=\int \sin t d t$ = -cos t + C = -cos(tan-1x) + C

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