Integration by Parts: Showing $\int x^3 e^x^2 dx = e^x^2 ( \frac{ x^2 -1}{ 2 })$

[Ed Aboud]

Homework Statement

Show using integration by parts that:

\int x^3 e^x^2 dx = e^x^2 ( \frac{ x^2 -1}{ 2 })

Homework Equations

The Attempt at a Solution

Integration by parts obviously.

\int u dv = uv - \int v du

Let u = x^3 and dv = e^x^2 dx

\int x^3 e^x^2 dx = \frac{x^2 e^x^2}{ 2 } - \frac{3}{2} \int x e^x^2 dx

Now use integration by parts again on \int x e^x^2 dx

And I get :

\frac{e^x^2}{ 2 } - \frac{1}{2} \int \frac{1}{x} e^x^2 dx

This really leaves me no closer again because I have to use integration by parts again on

\int \frac{1}{x} e^x^2 dx

Any suggestions on what to do.
Thanks for the help.

 

[Dick]

You mean e^(x^2). Your superscript isn't coming out. On the first step, you can't pick dv=e^(x^2)*dx. You can't integrate that. I have no idea what you are doing after that. Try dv=x*e^(x^2)*dx and u=x^2 for a first step. When you get to xe^(x^2), don't do parts again. Do it by an easy u-substitution (the same one you used to integrate dv).

 

[gabbagabbahey]

Homework Statement

Show using integration by parts that:

\int x^3 e^x^2 dx = e^x^2 ( \frac{ x^2 -1}{ 2 })

Homework Equations

The Attempt at a Solution

Integration by parts obviously.

\int u dv = uv - \int v du

Let u = x^3 and dv = e^x^2 dx

I don't think this is a very good choice for your u and dv, because v=\int dv=\int e^{x^2} dx is not e^{x^2}...try a substitution of the form w=x^2 before applying integration by parts

 

[Ed Aboud]

Cool, I showed it.
Thanks for the help!