<s>[INST] Hint for solving a definite integral $\int_{-a}^{a}\frac{xdy}{(x^{2}+y^{2})^{\frac{3}{2}}}$ Can anyone provide a hint for solving this definite integral:
$\int_{-a}^{a}\frac{xdy}{(x^{2}+y^{2})^{\frac{3}{2}}}$
 [/INST] Another way, in this particular example, is to rewrite your expression as
$$
\begin{aligned}
\frac{x}{(x^2+y^2)^{3/2}}&=\frac{x^2+y^2-y^2}{x(x^2+y^2)^{3/2}}=\frac{1}{x(x^2+y^2)^{1/2}}-\frac{y^2}{x(x^2+y^2)^{3/2}}\\
&=\Bigl(\frac{d}{dy} y\Bigr)\cdot \frac{1}{x(x^2+y^2)^{1/2}}+ y\cdot \frac{d}{dy}\frac{1}{x(x^2+y^2)^{1/2}}\\
&=\frac{d}{dy}\Bigl(\frac{y}{x(x^2+y^2)^{1/2}}\Bigr)
\end{aligned}
$$
Thus, you have a simple primitive function that you can put into the fundamental theorem of calculus.
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