Question

antiderivative of x[ln(x+3) ]

Answer 1

Integrate by parts, setting
\[\begin{matrix}
u=\ln(x+3)&&&dv=x~dx\\
du=\frac{dx}{x+3}&&&v=\frac{1}{2}x^2
\end{matrix}\]
Then
\[\int x\ln(x+3)~dx=\frac{1}{2}x^2\ln(x+3)-\frac{1}{2}\int\frac{x^2}{x+3}~dx\]
For the remaining, integral, I'd suggest a substitution, like \(t=x+3\), or long division.