Find e^ x ( ^ -1 x+1 1+x^ 2 ) d x - Vidyadip

Question

Find $\int e^{x}\left(\tan ^{-1} x+\frac{1}{1+x^{2}}\right) d x$

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Answer

We have $I=\int e^{x}\left(\tan ^{-1} x+\frac{1}{1+x^{2}}\right) d x$
Consider $f(x) = \tan^{-1} x,$ then $f′(x) = \frac{1}{1+x^{2}}$
Thus, the given integrand is of the form $e^x [f(x) + f′(x)].$
Therefore, $I=\int e^{x}\left(\tan ^{-1} x+\frac{1}{1+x^{2}}\right) d x$
$= e^x \tan^{-1} x + C$

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