What trick was used to transform this integral?

Question

I found this transformation in the solution set for some integration exercises. It looks like substitution, but it's written so I don't understand what the substitution was. $$\int_0^1 \frac{-2x_2^2}{(x_1^2+x_2^2)^2}dx_2 = \int_0^1x_2\frac{d}{dx_2}\left(\frac{1}{x_1^2+x_2^2}\right)dx_2$$ I'm confused: why is there a differentiation in the integral? What is happening here?

Answer 1

$\newcommand{\d}[1]{\; \mathrm{d} #1}$ More accurately it is a partial derivative. There's no substitution here - just evaluate the derivative on the right. $$ \int_0^1 x_2 \frac{\partial}{\partial x_2} \left(\frac{1}{x_1^2 + x_2^2}\right) \d{x_2} = \int_0^1 x_2 \left(\frac{-2x_2}{(x_1^2 + x_2^2)^2}\right) \d{x_2} = \int_0^1 \frac{-2x_2^2}{(x_1^2 + x_2^2)^2} \d{x_2} $$