Answer
$$\int x^3e^xdx=x^3e^x-3x^2e^x+6xe^x-6e^x+C$$
Work Step by Step
$$A=\int x^3e^xdx$$
Set $u=x^3$ and $dv=e^xdx$
Then we have $du=3x^2dx$ and $v=e^x$
Using the formula $\int udv= uv-\int vdu$:
$$A=x^3e^x-3\int x^2e^xdx$$
Set $u=x^2$ and $dv=e^xdx$
Then we have $du=2xdx$ and $v=e^x$
Using the formula $\int udv= uv-\int vdu$:
$$A=x^3e^x-3\Big(x^2e^x-2\int xe^xdx\Big)$$
Set $u=x$ and $dv=e^xdx$
Then we have $du=dx$ and $v=e^x$
Using the formula $\int udv= uv-\int vdu$:
$$A=x^3e^x-3\Bigg(x^2e^x-2\Big(xe^x-\int e^xdx\Big)\Bigg)$$ $$A=x^3e^x-3\Bigg(x^2e^x-2\Big(xe^x-e^x\Big)\Bigg)+C$$ $$A=x^3e^x-3\Bigg(x^2e^x-2xe^x+2e^x\Bigg)+C$$ $$A=x^3e^x-3x^2e^x+6xe^x-6e^x+C$$
