S3E33. Recurrence relations evaluate integrals

In this episode we explore a new to us approach for evaluation of integrals, a recurrence relation. In particular, using the given integral Sm as the fodder, we sculpt it into a relation that ties together that integral itself, Sm, and its lower-powered brother, the integral S(m--2), into a little engine, a recurrence relation, which, amazingly enough, can be solved by-hand. Using exactly the same approach, we do the same with the integral Cm. However, once the integral Sm is evaluated, we show that the like integral Cm does not really have to be crunched through because, according to a quite straightforward theorem, which we prove, the values of such integrals must match! In other words, we show that so long that a function F(u) is continuous in u over the subset of real numbers 0 to pi/2, it is the case that the value of the integral of F(sin(x)) over that range is exactly the value of the integral of F(cos(x)) over the same range. Thus, if the value of one such integral is known then so is the value of its complimentary cousin. As an extra for experts, at 30:21 and at 34:45, we consider 2 straightforward corollaries of the developed results. But our viewers can go even further: try to extend these results for the case when m is negative. The big arch that spans this and a couple of other episodes will support yet another episode in which we will show exactly what mathematicians do all day. For now we will leave it at that, a mysterious promise :O)