- #1

Ted123

- 446

- 0

## Homework Statement

## Homework Equations

For part (a):

## The Attempt at a Solution

I think I've done all of this question except for the very last part of (c).

For (a) I've found the integral [tex]\int \frac{x^2-y^2}{(x^2+y^2)^2}\;dx[/tex] using Wolfram Alpha but how would I integrate it bare handedly?

Since the double integral wrt x then y does not equal the double integral wrt y then x, f is not Lebesgue integrable over [0,1]x[0,1]. However if 0<a<b then f is Lebesgue integrable over [a,b]x[a,b] and the integral is 0 by Fubini.

The last part of (c) is what I'm not sure about: deducing that [tex]\int f = |S^+| - |S^-|.[/tex] Am I going about it the right way? We can write [itex]f=f^+ - f^-[/itex] where [itex]f^+ = \max (f,0)[/itex] and [itex]f^- = -\min (f,0)[/itex] (i.e. the +ve and -ve parts of f respectively).

Then we know if [itex]f\in L^1 (\mathbb{R})[/itex] then [itex]f^+ , f^- \in L^1 (\mathbb{R})[/itex] and [itex]f^+ , f^- \geq 0[/itex].

Now is the following correct? [tex]\int f = \int (f^+ - f^-) = \int f^+ - \int f^- = |S^+| - |S^-|[/tex]