Question

How do you find the antiderivative of `(cosx+secx)^2`?

Answer 1

`int \ (cosx+secx)^2 \ dx = sin(2x)/4 + 5/2x + tanx + C`

Explanation:

We seek:

`I = int \ (cosx+secx)^2 \ dx`

Which we can write:

`I = int \ cos^2x + 2cosxsecx + sec^2x \ dx`
`\ \ = int \ cos^2x + 2 + sec^2x \ dx`
`\ \ = int \ (cos(2x)+1)/2 + 2 + sec^2x \ dx`
`\ \ = int \ cos(2x)/2 + 5/2 + sec^2x \ dx`

All integrand functions are standard results, so we can now readily integrate to get:

`I = sin(2x)/4 + 5/2x + tanx + C`