Question

$\int e^x(ta{n}^{-1}x+\frac{1}{1+x^2})dx$

Answer 1

$\int e^x({\tan }^{-1}x+\frac{1}{1+x^2})dx$

As we know that,

$\int e^x(f\left(x\right)+f^{\prime }\left(x\right))dx=e^{x}f\left(x\right)+C$

$\because \frac{d}{dx}\left({\tan }^{-1}x\right)=\frac{1}{1+x^2}$

Therefore,

$\int e^x({\tan }^{-1}x+\frac{1}{1+x^2})dx=e^x{\tan }^{-1}x+C$