I'll have to check this alyosha. I've been lagging a little bit, trying to master the proof of the integral of cosine.. Off the top of my head, I think what he actually might be doing is using the concept of a piecewise monotonic function to construct the integral of a general polynomial. This is defended by that integration property for intervals, where the integral from a to b plus the integral from b to c is the integral from a to c. Sooner than not, I'll be eyeball deep in multivariable calculus and linear algebra. Think of the fun! Without even being grounded in the theoretical aspects of single variable calculus..