\(\int xln\left(x+1\right)dx\)
\(\left\{{}\begin{matrix}u=ln\left(x+1\right)\\dv=xdx\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}du=\dfrac{1}{x+1}dx\\v=\dfrac{x^2}{2}\end{matrix}\right.\)
\(\Rightarrow\int xln\left(x+1\right)dx=\dfrac{x^2}{2}.ln\left(x+1\right)-\int\dfrac{x^2}{2}.\dfrac{1}{x+1}dx=\dfrac{x^2}{2}.ln\left(x+1\right)-\dfrac{1}{2}\int\dfrac{x^2}{x+1}dx\)
Xet \(\int\dfrac{x^2}{x+1}dx=\int\dfrac{\left(x+1\right)\left(x-1\right)}{x+1}dx+\int\dfrac{1}{x+1}dx\)
\(=\int\left(x-1\right)dx+\int\dfrac{1}{x+1}dx\)
\(=\dfrac{x^2}{2}-x+ln\left(x+1\right)\)
\(\Rightarrow\int xln\left(x+1\right)dx=\dfrac{x^2}{2}.ln\left(x+1\right)-\dfrac{1}{2}\left(\dfrac{x^2}{2}-x+ln\left(x+1\right)\right)\)
