(x^2+2 ) x^2+1 a ^ x+ ^ -1 x d x - Vidyadip

Question

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

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Answer

Let $I =\int\left(\frac{x^2+2}{x^2+1}\right) a ^{x+\tan ^{-1 x}} d x$
Put $x+\tan ^{-1} x=t$
Differentiating w.r.t. $x$, we get
$ \left(1+\frac{1}{1+x^2}\right) d x= dt$
$\therefore\left(\frac{x^2+2}{x^2+1}\right) d x= dt$
$\therefore I =\int a ^1 dt$
$=\frac{ a ^1}{\log a }+ c$
$\therefore I =\frac{ a ^{x+\tan ^{-1 x}}}{\log a }+ c $

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