<s>[INST] How to integrate $\frac{y^2-x^2}{(y^2+x^2)^2}$ with respect to $y$? In dealing with the integration,
$$\int\frac{y^2-x^2}{(y^2+x^2)^2}dy$$
I have tried to transform it to polar form, which yields
$$\int\frac{\sin^2\theta-\cos^2\theta}{r^2}d(r\cos\theta)$$
But, what should I do now to continue?
I am sticking on it now.
 [/INST] $$
\int\frac{y^2-x^2}{(y^2+x^2)^2}dy
$$
\begin{align}
y & = x\tan\theta \\
dy & = x\sec^2\theta\,d\theta \\
y^2+x^2 & = x^2(\tan^2\theta+1) = x^2\sec^2\theta \\
y^2-x^2 & = x^2(\tan^2\theta-1) \\
& \phantom{=}\text{etc.}
\end{align}
Usually with $\displaystyle\int \Big(\cdots\cdots\text{something involving }(y^2+\text{constant}^2)\cdots\cdots\Big)\,dy$, this will work.
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