Thread: Help with an multiple integral

1. Help with an multiple integral

I'm a little rusty when it comes to these kind of integrals. Any help would be appreciated.

Find the indefinite integral of [(x^2 + y^2 - 2y^2)/(x^2 + y^2)^2]dx, that is, integrate with respect to x. I understand that I need to treat y as a constant, it's the integral itself that I'm having trouble with. Sorry if that's difficult to read, I still need to learn latex.

2. Originally Posted by spoon737
I'm a little rusty when it comes to these kind of integrals. Any help would be appreciated.

Find the indefinite integral of [(x^2 + y^2 - 2y^2)/(x^2 + y^2)^2]dx, that is, integrate with respect to x. I understand that I need to treat y as a constant, it's the integral itself that I'm having trouble with. Sorry if that's difficult to read, I still need to learn latex.
i believe you have a typo or something here. they would not say y^2 - 2y^2, they'd just write -y^2

3. Oops, you're right, that should read -y^2 instead of y^2 - 2y^2. This isn't how the actual problem was presented, I had to do some work to get to this point. I just forgot to simplify that one part. Still, I could use help in solving that integral.

4. Originally Posted by spoon737
Oops, you're right, that should read -y^2 instead of y^2 - 2y^2. This isn't how the actual problem was presented, I had to do some work to get to this point. I just forgot to simplify that one part. Still, I could use help in solving that integral.
Sorry for taking so long, here:

I will start you off (remember, we are treating y as a constant throughout the entire problem)

$\displaystyle \int \frac {x^2 - y^2}{(x^2 + y^2)^2} dx$

We employ the method of partial fractions:

$\displaystyle \frac {x^2 - y^2}{(x^2 + y^2)^2} = \frac {A}{x^2 + y^2} + \frac {B}{(x^2 + y^2)^2}$

$\displaystyle \Rightarrow x^2 - y^2 = A(x^2 + y^2) + B$ ...i think you know how to solve this, let's get to the meat of the matter

$\displaystyle \Rightarrow A = 1 \mbox { and } B = -2y^2$

$\displaystyle \Rightarrow \int \frac {x^2 - y^2}{(x^2 + y^2)^2}dx = \int \frac {1}{x^2 + y^2}dx + 2y^2 \int \frac {1}{(x^2 + y^2)^2}dx$

now just find the integrals on the right: factor out the $\displaystyle y^2$ on the bottom of the first so you will get it in the form to use arctan, for the second, use trig substitution, $\displaystyle x = ytan \theta$

If you get stuck you can check back with me