∫etan−1⁡x1+x+x21+x2dx is equal to

$\int e^{\tan^{- 1} x} (\frac{1 + x + x^{2}}{1 + x^{2}}) d x$ is equal to

A
$x e^{\tan^{- 1} x} + C$
B
$x^{2} e^{\tan^{- 1} x} + C$
C
$\frac{1}{x} e^{\tan^{- 1} x} + C$
D
none of these

Solution:

Putting $\tan^{- 1} x = t$ we have

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

$\Rightarrow l = \int e^{t} (\tan t + \sec^{2} t) d t = e^{t} \tan t + C = x e^{\tan^{- 1} x} + C$

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