I must solve $\displaystyle \displaystyle\int_{}^{}x|x|dx$
How should I proceed?
You can write $\displaystyle |x|=\sqrt{x^2}$. Then, you have $\displaystyle \int x|x|\text{ d}x=\int x\sqrt{x^2}\text{ d}x$. Let $\displaystyle u=x^2$ and $\displaystyle \text{d}u=2x\text{ d}x$. Your integral is $\displaystyle \frac{1}{2}\int \sqrt{u}\text{ d}u=\frac{1}{3}u^{3/2}+C=\frac{1}{3}(x^2)^{3/2}+C=\frac{1}{3}|x|^3+C$.