Consider the integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^3{e}^{-x^2}dx$ :

STEP $1$: First apply the $u$ substitution technique, with $u=x^2,$ to write ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^3{e}^{-x^2}dx$ in the form ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(u)du.$ Make sure you use the substitution $u=x^2$ and not anything else

Hint: the particular substitution $u=x^2$ makes sense if you write ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^3{e}^{-x^2}dx={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^2*e^{-x^2}*xdx$

${\displaystyle \underset{\text{w}\text{i}\text{t}\text{h}\text{v}\text{a}\text{r}\text{i}\text{a}\text{b}\text{l}\text{e}u}{\overset{\phantom{}}{}}}$

STEP $2$: Now, use integration by parts to compute the indefinite integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(u)du,$ which is the anti$-$derivative of $f(u).($Hint: You already did this integral $($without the coefficient of $\frac{1}{2})$ in a previous homework problem using variable $x$ instead of $u.$ Consult that problem.$)$

anti$-$derivative of $f(u)=+C($write in terms of variable $u)$

STEP $3$: Write you previous answer replacing $u$ with $x^2.$

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^3{e}^{-x^2}dx=,+C$