|Description: I am trying to find the indefinite integral of sqrt(tanx) dx and the indefinite integral of sqrt(cos2x) dx. Any ideas?
Idea 1: There seem to be no known closed form antiderivatives for the given functions (in terms of the elementary functions). However, the antiderivatives certainly exist; Expand the given integrand in terms of its Taylor series at x=0. Integrating the Taylor series term by term gives the Taylor series expansion of the antiderivative.
Idea 2: Usually, one is interested in computing the definite integral. For this purpose, there are many numerical methods known. For example, Simpson's Rule.