There is a very pretty theorem that says: If c is a positive real number and there exist a function f(x) that is continuous on [0,c], positive on (0,c) and such that f and it iterated anti-derivatives can be taken to be integer valued at 0 and c, then c is irrational!
pi is a positive real number. The function f(x)= sin(x) is continuous for all x and so on [0,pi] and is positive on (0, pi). All anti-derivatives can be taken to be +/- sin(x) or +/- cos(x) (by always choosing the constant of integration to be 0). Therefore pi is irrational.