How do you integrate int 1/sqrt(e^(2x)-2e^x-24)dx using trigonometric substitution?
1 Answer
Jun 11, 2018
Use the substitution
Explanation:
Let
I=int1/sqrt(e^(2x)-2e^x-24)dx
Rewrite in terms of
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
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