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!