Evaluate the following integrals:

Question:

Evaluate the following integrals:

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

Solution:

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

here, $\mathrm{f}(\mathrm{x})=\tan ^{-1} \mathrm{x}$ and $\mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{1+\mathrm{x}^{2}}$

and we know that,

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

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

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