Question

$\int \log \left(1+x^2\right)dx$

Answer 1

$$\begin{gathered} \int \underset{\mathrm{II}}{1}.\underset{\mathrm{I}}{\log \left(1+x^2\right)}dx \\ =\log \left(1+x^2\right)\int 1dx-\int \left\{\frac{d}{dx}\log \left(1+x^2\right)\int 1dx\right\}dx \\ =\log \left(1+x^2\right).x-\int \frac{2x}{1+x^2}.xdx \\ =x\log \left(1+x^2\right)-2\int \left(\frac{x^2+1-1}{1+x^2}\right)dx \\ =x\log \left(1+x^2\right)-2\int dx+2\int \frac{dx}{1+x^2} \\ =x\log \left(1+x^2\right)-2x+2{\tan }^{-1}\left(x\right)+C \end{gathered}$$