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

1 Answer
Jun 20, 2018

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