Question

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

Answer 1

$$\begin{gathered} \int \log \left(x+1\right)dx \\ =\int 1.\log \left(x+1\right)dx \\ \mathrm{Taking}\log \left(x+1\right)\mathrm{as}\mathrm{the}\mathrm{first}\mathrm{function}\mathrm{and}1\mathrm{as}\mathrm{the}\mathrm{second}\mathrm{function}. \\ =\log \left(x+1\right)\int 1dx-\int \left[\frac{d}{dx}\left\{\log \left(x+1\right)\right\}\int 1dx\right]dx \\ =x\log \left(x+1\right)-\int \frac{x}{x+1}dx \\ =x\log \left(x+1\right)-\int \frac{x+1}{x+1}-\frac{1}{x+1}dx \\ =x\log \left(x+1\right)-x+\log \left|x+1\right|+C \end{gathered}$$