# Integration by Parts

#### paulmdrdo

##### Active member
any idea how to solve this?

\begin{align*}\displaystyle \int x^3e^{x^{2}}\,dx\end{align*}

##### Well-known member
Re: integration by parts

any idea how to solve this?

\begin{align*}\displaystyle \int x^3e^{x^{2}}\,dx\end{align*}
put x^2 = t

so 2 x dx = dt

so x^3 e^(x^2) dx = t e^t dt / 2

now you can differentiate t and integrate e^t thus by parts.

#### soroban

##### Well-known member
Re: integration by parts

Hello, paulmdrdo!

$$I \;=\; \int x^3e^{x^2}\,dx$$

We have: .$$\int x^2\cdot xe^{x^2}dx$$

By parts: .$$\begin{Bmatrix}u &=& x^2 && dv &=& xe^{x^2}dx \\ du &=& 2x\,dx && v &=& \tfrac{1}{2}e^{x^2} \end{Bmatrix}$$

Then: .$$I \;=\;\tfrac{1}{2}x^2e^{x^2} - \int xe^{x^2}dx$$

. . . . . $$I \;=\;\tfrac{1}{2}x^2e^{x^2} - \tfrac{1}{2}e^{x^2} + C$$

. . . . . $$I \;=\;\tfrac{1}{2}e^{x^2}(x^2-1) + C$$