Question

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

Answer 1

$$\begin{gathered} \int 5^{x+{\tan }^{-1}x}·\left(\frac{x^2+2}{x^2+1}\right)dx \\ \mathrm{Let}x+{\tan }^{-1}x=t \\ \left(1+\frac{1}{1+x^2}\right)=\frac{dt}{dx} \\ \Rightarrow \left(\frac{x^2+1+1}{x^2+1}\right)dx=dt \\ \Rightarrow \left(\frac{x^2+2}{x^2+1}\right)dx=dt \\ \mathrm{Now},\int 5^{x+{\tan }^{-1}x}·\left(\frac{x^2+2}{x^2+1}\right)dx \\ =\int 5^{t}dt \\ =\frac{5^t}{\log 5}+\mathrm{C} \\ =\frac{5^{x+{\tan }^{1}x}}{\log 5}+\mathrm{C} \end{gathered}$$