# Substitution and integration by parts

## Homework Statement

Can anybody help me integrate $$x^3 e^{x^2}$$

## The Attempt at a Solution

I can't see how to do it by substitution or integration by parts.

## Answers and Replies

G01
Homework Helper
Gold Member
Here's a hint. Did you try doing one and THEN the other?

## Homework Statement

Can anybody help me integrate $$x^3 e^{x^2}$$

## The Attempt at a Solution

I can't see how to do it by substitution or integration by parts.

Integration by parts should work. Set u equal to the term which will diminish eventually as you keep taking the derivative of and dv equal to the other term.

Avodyne
Science Advisor
It's hard to integrate e^(x^2). It's easy to integrate e^x. Does this suggest a substitution you might consider?

CompuChip
Science Advisor
Homework Helper
Actually exp(x^2) is a sort of standard integral. It's called a Gaussian integral. You can find it in about any physics book, vector calculus book or at the front of any integral table, sometimes with proof. The idea is basically to calculate
$$\left( \int e^{x^2} \, dx \right)^2 = \left( \int e^{x^2} \, dx \right) \left( \int e^{y^2} \, dy \right) = \left( \int e^{x^2 + y^2} \, dx \, dy \right)$$
and evaluate it using polar coordinates.

Avodyne
Science Advisor
That only works if you're doing a definite integral over an infinite range. The question asks for the indefinite integral.

dextercioby
Science Advisor
Homework Helper
Not to mention that you need a "-" sign in the exponent, else it won't converge...

HallsofIvy
Science Advisor
Homework Helper
Avodyne's suggestion is the best in my opinion.

I think I see where I messed up in my substitution, which made me think it'd be useless

So with a u substitution I should get $$\frac{1}{2}u \cdot e^{u}$$ ?

Thanks for all the suggestions.

Last edited:
dynamicsolo
Homework Helper
I think I see where I messed up in my substitution, which me me think it'd be useless

So with a u substitution I should get $$\frac{1}{2}u \cdot e^{u}$$ ?

Thanks for all the suggestions.

Yes, that does work. (I was still editing my reply while you posted this.) You should get

(1/2)[ (x^2) - 1 ] · exp(x^2) + C .

It's one of those oddball integrals that looks like one of the exponents went the wrong way, but when you check your result, differentiating the exp(x^2) provides the factor that gets you the (x^3) ...

Last edited:
HallsofIvy
Science Advisor
Homework Helper
Actually, you should get
$$\frac{1}{2}u e^u du$$