[tex] \int xln(x+1)dx= \frac{1}{2} \int(x^2)'ln(x+1)dx= \\ =\frac{1}{2}x^2ln(x+1)- \frac{1}{2} \int x^2[ln(x+1)]'dx= \\ =\frac{1}{2}x^2ln(x+1)- \frac{1}{2} \int x^2 \frac{1}{x+1} dx=
\\ =\frac{1}{2}x^2ln(x+1)- \frac{1}{2} \int x-1+\frac{1}{x+1}dx= \\
=\frac{1}{2}x^2ln(x+1)- \frac{1}{2} ( \frac{x^2}{2}-x+ln|x+1|)= \\
=\frac{1}{2}x^2ln(x+1)- \frac{x^2}{4}+\frac{x}{2}-\frac{ln|x+1|)}{2} [/tex]
