# 022 Sample Final A, Problem 12

Find the antiderivative: ${\displaystyle \int x^{2}e^{3x^{3}}dx.}$

Foundations:
This problem requires an advanced rule of integration, namely
Integration by substitution (u - sub): If ${\displaystyle u=g(x)}$  is a differentiable functions whose range is in the domain of ${\displaystyle f}$, then
${\displaystyle \int g'(x)f(g(x))dx\,=\,\int f(u)du.}$

Solution:

Step 1:
Use a ${\displaystyle u}$-substitution with ${\displaystyle u=3x^{3}.}$ This means ${\displaystyle du=9x^{2}\,dx}$, or  ${\displaystyle {\frac {du}{9x^{2}}}\,=\,dx}$. After substitution, we have
${\displaystyle \int x^{2}e^{3x^{3}}dx\,=\,\int x^{2}e^{u}\cdot {\frac {du}{9x^{2}}}\,=\,{\frac {1}{9}}\int e^{u}\,du.}$
Step 2:
From what should be well-known property,
${\displaystyle {\frac {1}{9}}\int e^{u}\,du\,=\,{\frac {1}{9}}\,e^{u}.}$
Step 3:
Now we need to substitute back into our original variables using our original substitution ${\displaystyle u=3x^{3}}$ to find  ${\displaystyle e^{u}\,=\,e^{3x^{3}}}$.
Step 4:
Since this integral is an indefinite integral, we have to remember to add a constant  ${\displaystyle C}$ at the end.
${\displaystyle \int x^{2}e^{3x^{3}}dx\,=\,{\frac {1}{9}}\,e^{3x^{3}}+C.}$