# How do you integrate int x^2e^(x^3) by parts?

Dec 18, 2016

Using integration by parts is very artificial for this integral. Substitution is much more reasonable.

#### Explanation:

$\int {x}^{2} {e}^{{x}^{3}} \mathrm{dx}$

Let $u = {x}^{3}$. This makes $\mathrm{du} = 3 {x}^{2} \mathrm{dx}$.

The integral becomes

$\frac{1}{3} \int {e}^{{x}^{3}} \left(3 {x}^{2} \mathrm{dx}\right) = \frac{1}{3} \int {e}^{u} \mathrm{du}$

$= \frac{1}{3} {e}^{u} + C$

$= \frac{1}{3} {e}^{{x}^{3}} + C$

If I am told that I must use parts ,

I'll let $u = 1$ and $\mathrm{dv} = {x}^{2} {e}^{{x}^{3}} \mathrm{dx}$

so that $\mathrm{du} = 0 \mathrm{dx}$ and $v = \frac{1}{3} {e}^{{x}^{3}}$.

And

$u v = \int v \mathrm{du} = 1 \cdot \frac{1}{3} {e}^{{x}^{3}} - \int \frac{1}{3} {e}^{{x}^{3}} \cdot 0 \mathrm{du}$

$= \frac{1}{3} {e}^{{x}^{3}} + C$.