Question:
Evaluate $\int \frac{\mathrm{e}^{\mathrm{m} \tan ^{-1} \mathrm{x}}}{\left(1+\mathrm{x}^{2}\right)^{3 / 2}} \mathrm{dx}$
Solution:
$=e^{m} \int \frac{\tan ^{-1} x}{\left(1+x^{2}\right) \sqrt{1+x^{2}}} d x$
Put $\tan ^{-1} x=t, d x /\left(1+x^{2}\right)=d t, 1+x^{2}=\sec ^{2} x$;
$=e^{m} \int \frac{t d t}{\sec t}=e^{m} \int t \cos t d t$
$=e^{m}\left[t \sin t-\int \sin t d t\right]$
$=e^{m}[t \sin t+\cos t]+c$
$=e^{m}\left[\frac{x \tan ^{-1} x}{\sqrt{1+x^{2}}}+\frac{1}{\sqrt{1+x^{2}}}\right]+c$
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