How do you integrate int 1/sqrt(e^(2x)-2e^x-24)dx using trigonometric substitution?

1 Answer
Jun 11, 2018

Use the substitution 24e^-x+1=5sintheta.

Explanation:

Let

I=int1/sqrt(e^(2x)-2e^x-24)dx

Rewrite in terms of e^-x:

I=inte^-x/sqrt(1-2e^-x-24e^(-2x))dx

Complete the square in the square root:

I=sqrt24inte^-x/sqrt(25-(24e^-x+1)^2)dx

Apply the substitution 24e^-x+1=5sintheta:

I=sqrt24int1/(5costheta)(-5/24costhetad theta)

Simplify:

I=-1/sqrt24intd theta

The integral is trivial:

I=-1/sqrt24theta+C

Reverse the substitution:

I=-1/sqrt24sin^(-1)((24e^-x+1)/5)+C