\int _0^xf'(x)\text{dx}=\int _0^x\left(\sum _{n=0}^{\infty } (-1)^nx^{2n}\right)\text{dx}=\sum _{n=0}^{\infty } (-1)^n\int _0^xx^{2n}\text{dx}=\sum _{n=0}^{\infty } \frac{(-1)^nx^{2n+1}}{2n+1}