Answer
The statement is False.
Work Step by Step
The statement is False since this integral
$$
\int x^{3}e^{-x^{2}}dx
$$
Apply the method of integration by parts only one time to determine the given integration:
Let $u=x^{2}$ and $dv =xe^{-x^{2}}dx$, so that:
$$
\begin{aligned} \int x^{3}e^{-x^{2}}dx &= \int x^{2}. xe^{-x^{2}}dx\\
&= -\frac{1}{2} \int x^{2}. (-2xe^{-x^{2}}dx)\\
&= -\frac{1}{2} \int x^{2}. d(e^{-x^{2}})\\
&\quad \left[\begin{array}{c}{u=x^{2}, \quad\quad dv= -2xe^{-x^{2}}dx} \\ {d u= 2xdx , \quad\quad v=e^{-x^{2}} }\end{array}\right] , \text { then }\\
&= -\frac{1}{2} [x^{2} e^{-x^{2}} -2\int x. e^{-x^{2}}dx]\\
&= -\frac{1}{2} [x^{2} e^{-x^{2}} +e^{-x^{2}}].\\
\end{aligned}
$$
Therefore, the statement is False.
