MHB Solving Integration by Parts: x^3e^x^2

[paulmdrdo1]

any idea how to solve this?

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

 

[kaliprasad]

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]

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