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I'm self-studying a Cramster solution and I came across this integral and I don't know what they've done with it. Help would be appreciated.

$\int \frac{y^2 - x^2}{(x^2 + y^2)^2} ~dy.$

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    Is x constant wrt y?2012-07-17

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In an integral $dy$, $x$ is a constant. Rewirite this as: $\int\frac{y^2-x^2}{\left(x^2+y^2\right)^2}dy=\int\frac{y^2+x^2-2x^2}{\left(x^2+y^2\right)^2}dy=\int\frac{dy}{x^2+y^2}-2x^2\int\frac{dy}{\left(x^2+y^2\right)^2}$ Can you continue from here?

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    Yes, this is a simple example of the partial fractions method.2012-07-17
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If x is constant wrt y,

let $y=x. tan(z)$, then $dy=xsec^2zdz $

Then $\int \frac{y^2 - x^2}{(x^2 + y^2)^2} ~dy$ becomes

$\frac{-1}{x}\int cos(2z) ~dz$

=$\frac{-sin(2z)}{2x} + C$, where C is an undetermined constant.

=$\frac{-tan(z)}{x(1+tan^2z)} +C$

=$\frac{-y}{x^2+y^2}+ C$