Question

How do you integrate `int 1/sqrt(1-4x^2)` by trigonometric substitution?

Answer 1

`int 1/(sqrt(1-4x^2)) dx = 1/2 sin^(-1) (2x) + C`

Explanation:

Let `x = 1/2 sin theta`

Then:

`int 1/(sqrt(1-4x^2)) dx = int 1/sqrt(1 - sin^2 theta) (d 1/2 sin theta)/(d theta) d theta`

`color(white)(int 1/(sqrt(1-4x^2)) dx) = int (1/2 cos theta)/cos theta d theta`

`color(white)(int 1/(sqrt(1-4x^2)) dx) = int 1/2 d theta`

`color(white)(int 1/(sqrt(1-4x^2)) dx) = 1/2 theta + C`

`color(white)(int 1/(sqrt(1-4x^2)) dx) = 1/2 sin^(-1) (2x) + C`