Best way to integrate a moment generating function?

[trojansc82]

Homework Statement

∫e^txx²e^-x

Homework Equations

M(t) = e^tx f(x) dx

The Attempt at a Solution

I know the solution is -1/(t-3)³, however I'm having difficulty integrating the function. UV - ∫ V DU is extremely long and challenging, I'm wondering if there is a shortcut (i.e. quotient rule?)

Also, there is a process used here but I'm unable to understand it:

 

[LCKurtz]

Homework Statement

∫e^txx²e^-x

Homework Equations

M(t) = e^tx f(x) dx

The Attempt at a Solution

I know the solution is -1/(t-3)³, however I'm having difficulty integrating the function. UV - ∫ V DU is extremely long and challenging, I'm wondering if there is a shortcut (i.e. quotient rule?)

Unfortunately, integration by parts is the antiderivative method generated by the product or quotient rules. I don't know of any shortcut. Let u = x² the first round than u = x the second round and you should be able to integrate the result directly.

 

[lurflurf]

It looks like that example integrated by parts multiple times. It is easy if you practice. Another method could be differentiation by the parameter.
Let I_n(t) be integrals like yours where n is the power of x

Then notice
I_n(t)=DⁿI₀(t)=Dⁿ(-1/(t-1))

where
D is differentiation with respect to t
I₀(t)=(-1/(t-1))

 

[trojansc82]

Unfortunately, integration by parts is the antiderivative method generated by the product or quotient rules. I don't know of any shortcut. Let u = x² the first round than u = x the second round and you should be able to integrate the result directly.

I did the integration by parts, and I ended up with 1/(1-t)³...is that incorrect?

The mean and variance were still the same as the book's answers (μ = 3, σ² = 3)