Question

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

Answer 1

`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`