Question:
Evaluate the following integrals:
$\int 5^{x+\tan ^{-1} x}\left(\frac{x^{2}+2}{x^{2}+1}\right) d x$
Solution:
Assume $x+\tan ^{-1} x=t$]
$d\left(x+\tan ^{-1} x\right)=d t$
$\Rightarrow 1+\frac{1}{x^{2}+1}=d t$
$\Rightarrow \frac{2+\mathrm{x}^{2}}{\mathrm{x}^{2}+1}=\mathrm{dt}$
Substituting $\mathrm{t}$ and $\mathrm{dt}$
$\Rightarrow \int 5^{t} d t$
$\Rightarrow \frac{5^{t}}{\log 5}+c$
But $\mathrm{t}=\mathrm{x}+\tan ^{-1} \mathrm{x}$
$\Rightarrow \frac{5^{x+\tan ^{-1} x}}{\log 5}+c$
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