Question

How do you integrate `int dx/(4x^2-1)^(3/2)` using trig substitutions?

Answer 1

Use the substitution `2x=sectheta`.

Explanation:

Let

`I=intdx/(4x^2-1)^(3/2)`

Apply the substitution `2x=sectheta`:

`I=int(1/2secthetatanthetad theta)/(tan^3theta)`

Simplify:

`I=1/2intcscthetacotthetad theta`

Integrate directly:

`I=-1/2csctheta+C`

Rewrite in terms of `sectheta` and `tantheta`:

`I=-1/2 sectheta/tantheta+C`

Reverse the substitution:

`I=-x/sqrt(4x^2-1)+C`