022 Sample Final A, Problem 12

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

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

 Solution:

Step 1:
Use a $u$-substitution with $u=3x^{3}.$ This means $du=9x^{2}\,dx$, or  ${\frac {du}{9x^{2}}}\,=\,dx$. After substitution, we have $\int x^{2}e^{3x^{3}}dx\,=\,\int x^{2}e^{u}\cdot {\frac {du}{9x^{2}}}\,=\,{\frac {1}{9}}\int e^{u}\,du.$

Step 2:
From what should be well-known property, ${\frac {1}{9}}\int e^{u}\,du\,=\,{\frac {1}{9}}\,e^{u}.$

Step 3:
Now we need to substitute back into our original variables using our original substitution $u=3x^{3}$ to find  $e^{u}\,=\,e^{3x^{3}}$.

Step 4:
Since this integral is an indefinite integral, we have to remember to add a constant  $C$ at the end.

Final Answer:
$\int x^{2}e^{3x^{3}}dx\,=\,{\frac {1}{9}}\,e^{3x^{3}}+C.$

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