Double Integral Problem: How to Evaluate ∫1 to 4∫0 to y(2/(x^2+y^2))dxdy

[aglo6509]

Homework Statement

Evaluate: ∫1 to 4∫0 to y(2/(x^2+y^2))dxdy

Homework Equations

The Attempt at a Solution

So I know you have to spilt it up and do the dx integral first:

∫0-y(2/(x^2+y^2))dx

Now this is where I don't know if I'm doing it right, I moved the 2 outside the integral and split up the fraction, so:

2(∫1/x^2dx+∫1/y^2dx)

Now since I'm only dealing with dx I'll ignore the y for right now:

∫1/x^2= -1/x|0to y
= -1/y

So the new integral is:

∫-2/(y+y^2)dy

Again move the two outside and split up the intgeral:

-2(∫1/ydy-∫1/y^2dy)
-2(lny+1/y^2)from 1 to 4

then it's just imputing numbers.

So basically if you could tell me if I'm right about being able to split up the fraction like I do I'd very much appreciate it!

 

[SammyS]

$\displaystyle\frac{1}{x^2+y^2}\ne\frac{1}{x^2}+ \frac{1}{y^2}$

Treat y as a constant when integrating with respect to x.

$\displaystyle \int\frac{1}{x^2+a^2}\,dx=\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C$