# Thread: [SOLVED] Integration by Parts

1. ## [SOLVED] Integration by Parts

I've having trouble integrating this expression by parts:

The first expression is the one I want to solve by using the second equation.

I've tried solving it by parts but I can't seem to impliment the limits.

Just so you know, the answer is:

4[((llamda^3)/2*Pi)]^(1/4)

2. Hello,

Write $\int_0^\infty x \cdot \left(x e^{-2\lambda x^2}\right) ~dx$

You can see that x is (with some constant) the derivative of $-2\lambda x^2$

So let $u=x$ and $dv=xe^{-2\lambda x^2}$

3. Ah cheers, that's nifty.

What do I do if it's x^3 in front of the integral instead of x^2?

[Edit: Nevermind just worked it out, seems obvious now ]

4. And here comes the next inevitable question.

What happens if it's x^4 in front of the exponential expression? Is there a similar easy trick?

5. Originally Posted by Gingerbread Man
And here comes the next inevitable question.

What happens if it's x^4 in front of the exponential expression? Is there a similar easy trick?
Yup, isolating a factor x to get $xe^{-2\lambda x^2}$ and then integration by parts, successively

If you want another method... : you can recognize the Gamma function (http://en.wikipedia.org/wiki/Gamma_function), by letting $t=-2\lambda x^2$.

6. Thankyou!