Find $\int x\ln(x^2e^{x^2})\,dx.$ How can I find the following integral $$\int x\ln(x^2e^{x^2})\,dx.$$ Which substitution may I use to solve the integral?

Note that by applying logarithms addition rule you get
$$\begin{align}
\int x\ln(x^2e^{x^2})dx &= \int x\left(\ln(x^2) + \ln(e^{x^2})\right)dx\\
&=\int x\left(2\ln x  + x^2\right)dx\\
&=\int 2x\ln x + x^3dx\\
&= 2\int x\ln x + \int x^3dx\\
&= 2\left( \frac{1}{2} x^2 \log (x)-\frac{x^2}{4} \right) + \frac{x^4}{4} + C\\
&= x^2 \log x - \frac{x^2}{2} + \frac{x^4}{4}
\end{align}$$
