# Integration Partial Fractions

## Main Question or Discussion Point

Im reading Lang's first course in calculus and can't understand one step that he does when trying to integrate quotients with quadratic factors in the denominator. He's trying to find the integral of $$\int{\frac{1}{(x^2+1)^n}dx}$$

but he's first starting with the case where n=1

Then while using integration by parts he gets this integral for $\int{vdu}$ $$2 \int{\frac{x^2}{(x^2+1)^2}dx}$$

then he writes x^2=x^2+1-1 and gets $$\int{\frac{1}{x^2+1}dx}-\int{\frac{1}{(x^2+1)^2}dx}$$

now I dont understand why he gets those two integrals as a result of writing x^2 as x^2+1-1, can anybody please help me out here?

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