Substitution and then integration by parts

[MarkSA]

Hello,

I have a couple of problems that state:
"First make a substitution, and then use integration by parts to evaluate the integral."

My book has no examples on doing this (the 'first make a substitution bit'.. I can integrate by parts) so i'm totally lost about how to do this, or even what they mean by 'make a substitution.' Do you have any idea what they mean?

One problem for this is:

1) Find the integral of: x^(5)e^(x^2)dx

Thanks

 

[o_O]

It means you're going have to perform a u substitution which will result in taking the integral of a product. To start you off:

\(\displaystyle u = x^{2} \quad du = 2xdx\)

And imagine:

\(\displaystyle \int x^{5} e^{x^{2}} dx = \int x \cdot x^{4} \cdot e^{x^{2}} dx\)

 

[MarkSA]

Sorry i'm still not seeing where to begin... I can see why those two integrals are equal, but not the purpose of the u-sub for x^2 nor what to do with it.

You mentioend taking the integral of a product, but the original integral is already an integral of a product?

 

[o_O]

Well from where I left off:

\(\displaystyle u = x^{2}\)

\(\displaystyle \frac{du}{2} = x dx\)

So:

\(\displaystyle \int x^{5} e^{x^{2}}dx = \int x^{4} e^{x^{2}} \underbrace{x dx}_{du/2} = \int (x^{2})^{2} e^{x^{2}} x dx\)

See where I'm going with this?

 

[MarkSA]

Yes I think I see it now.

integral of: u^(2)e^(u)du

and integration by parts should be doable... i'll try it later thanks. So I should just have to use v/dv and w/dw or another variable when i do integration by parts since u is taken?

So the purpose of the substitution is to make integration by parts easier, is that correct?

 

[o_O]

Pretty much. And don't forget the \(\displaystyle \frac{1}{2}\) in front of the integral since:

\(\displaystyle \underbrace{\frac{1}{2}}du = xdx\)