Efficient Integration of x^5 exp(x^2) in First-Year Calculus

[kingwinner]

Homework Statement

I am losing my first year calculus skills :(
I don't remember how to integrate x⁵ exp(x²).
What is the fastest way?

Homework Equations

N/A

The Attempt at a Solution

Maybe we need to integrate by parts? But how should I set u and dv?


Thanks for any help!

 

[Mark44]

I would go with integration by parts. There are various possibilities for u and dv, but the one I would try first is u = x⁴, dv = xe^x2dx. A good strategy for integration by parts is to choose dv so that it is the most complicated thing that you can actually integrate.

The goal is to get an integral with x to a power less than 5, and keep applying integration by parts until you get a fairly simple integral, like $$\int xe^{x^2}dx$$, which can be done by an ordinary substitution.

 

[Brian-san]

If you are familiar with the trick of tabular integration for integration by parts, you can use the substitution u=x², du=2xdx, to get the solution faster without having to perform multiple integrations by parts.

 

[kingwinner]

How many times do I have to integrate by parts?

 

[Brian-san]

Depending on the substitution used, at least 2 times.

 

[kingwinner]

This integral actually occurs in the middle of a statistics problem. If I know the expectation of a gamma distribution, can I possibly avoid integrating by parts in the above integral? If so, how?

 

[Mark44]

This integral actually occurs in the middle of a statistics problem. If I know the expectation of a gamma distribution, can I possibly avoid integrating by parts in the above integral? If so, how?

I don't know anything about that. You posed the problem, and you have gotten a couple of strategies for solving it. If you don't know how to do integration by parts, say so, and we'll help you out.

 

[g_edgar]

Another method that works here is to substitute u=x^2 ...
This will work for the integral of x^n exp(x^2) when n is odd.