We will outline an efficient algorithmic approach to the desingularisation of plane algebraic curves. Applications include computing the genus, Riemann-Roch spaces, and testing whether the curve is hyperelliptic. Afterwards, we will see that the (apparently rustic-looking) problem of finding the antiderivative of an algebraic function is actually related to the (much cooler-sounding) ability to test whether certain divisors are torsion in the Picard group of a curve. We will show how to achieve this thanks to the algorithms outlined earlier, which will lead us to a complete integration algorithm for algebraic functions based on arithmetic geometry. The talk will feature many explicit examples.