The series (of continuous functions) converges uniformly, which means that it's Reimann integrable. And you can compute its integral as a sum of the integrals

The series (of continuous functions) converges uniformly, which means that it's Reimann integrable. And you can compute its integral as a sum of the integrals inside the sum (which you showed upthread). Also, as hinted upthread, just change f(x) = - sum[n = 0 to infinity](10^n / pi cos((1/20)^n pi x)) to f(x) = - sum[n = 0 to infinity](10^n / pi (cos((1/20)^n pi x) - 1)) That gives you an actual formula for the antiderivative that converges at every point.