Answer
$$\int x^5e^{x^3}dx=\frac{1}{3}(x^3e^{x^3}-e^{x^3})+C$$
Work Step by Step
$$A=\int x^5e^{x^3}dx=\int x^3e^{x^3}(x^2dx)$$
We set $a=x^3$. We then have $$da=3x^2dx$$ $$x^2dx=\frac{1}{3}da$$
Therefore, $$A=\frac{1}{3}\int ae^ada$$
Next, take $u=a$ and $dv=e^ada$
We would have $du=da$ and $v=e^a$
Applying the formula $\int udv=uv-\int vdu$, we have $$A=\frac{1}{3}(ae^a-\int e^ada)$$ $$A=\frac{1}{3}(ae^a-e^a)+C$$ $$A=\frac{1}{3}(x^3e^{x^3}-e^{x^3})+C$$
