How do you find the antiderivative of #sqrt cosx#?

1 Answer
Oct 21, 2016

#2E(x/2 | 2)+C#

Explanation:

#I=intsqrtcos(x)dx#

Using the cosine double angle formula #cos(2x)=1-2sin^2(x)#, we also see that #cos(x)=1-2sin^2(x/2)#:

#I=sqrt(1-2sin^2(x/2))dx#

Letting #u=x/2# and #du=1/2dx#:

#I=2intsqrt(1-2sin^2(u))du#

This is a special integral, namely the incomplete elliptic of the second kind #E#. It is defined by #E(varphi|k^2)=int_0^varphisqrt(1-k^2sin^2(theta))d theta#.

So, here, we see that:

#I=2E(u|2)+C=2E(x/2 | 2)+C#