Question

How do you find the antiderivative of `cos^(-1)x dx`?

Answer 1

Use parts, or substitution and then parts.

Explanation:

`int cos^-1 x dx`

Let `u = cos^-1x` so `du = -1/(sqrt(1-x^2) dx` and

let `dv = dx`, so `v = x`

`uv-int vdu = xcos^-1x + int x/sqrt(1-x^2) dx`

Use substitution `u = 1-x^2` to finish with

`int cos^-1x dx = xcos^-1 x - sqrt(1-x^2) +C`

OR replace the inverse cosine first

Let `theta = cos^-1x` so that `cos theta = x` and -sin theta d theta = dx#

The integral becomes

`-inttheta sin theta d theta` Which may be integrated by parts with `u = theta` and `dv = sin theta d theta`

When finished we get

`- (sin theta - theta cos theta) + C`

And reversing the substitution gets `xcos^-1 x - sqrt(1-x^2) +C` as above.