Question

$\int 5^{x+{\tan }^{-1}x}\left(\frac{x^2+2}{x^2+1}\right)dx$

Answer 1

We have, $\frac{d}{dx}(x+{\tan }^{-1}x)=1+\frac{1}{1+x^2}=\frac{x^2+2}{x^2+1}$........(1)

$\int 5^{x+{\tan }^{-1}x}\left(\frac{x^2+2}{x^2+1}\right)dx$

$=\int 5^{x+{\tan }^{-1}x}d(x+ta{n}^{-1}x)$ [ Using (1)]

$=\frac{5^{x+{\tan }^{-1}x}}{\log 5}+c$ [ Where $c$ is integrating constant]