# trig/hyperbolic substitution integral

• Sep 7th 2013, 05:39 AM
tylersmith7690
trig/hyperbolic substitution integral
Using a trigonometric or hyperbolic substitution, evaluate the following inde nite integral,

∫ 1 / ((x^2)-1)^(5/2) dx

my attempt:

let x= cosh y , dx/dy = sinh y , dx = sinhy dy

then (x^2-1) = (cosh^2 y) -1
= sinh^2 y

so ((x^2)-1)^(1/2) = sinh y
((x^2)-1)^(5/2) = ( sinh y ) ^5

so integral becomes sinh y / (sinh y)^5 dy

which is 1/ ( sinh y)^4 or ( cosech y )^2

I have tried to simplify these two integrals, one by putting cosh^2 y - sinh ^2 y in the numerator to replace the one in 1/ (sinh y)^4. But couldn't seem to get anywhere with the results. Is there some clever trick to apply to where I got down. I have spent way to long on this question and would like to see how someone else would work it out and give me some insight to there thinking.

Kind regards.
• Sep 7th 2013, 05:44 AM
FelixFelicis28
Re: trig/hyperbolic substitution integral
Quote:

Originally Posted by tylersmith7690
Using a trigonometric or hyperbolic substitution, evaluate the following indenite integral,

∫ 1 / ((x^2)-1)^(5/2) dx

my attempt:

let x= cosh y , dx/dy = sinh y , dx = sinhy dy

then (x^2-1) = (cosh^2 y) -1
= sinh^2 y

so ((x^2)-1)^(1/2) = sinh y
((x^2)-1)^(5/2) = ( sinh y ) ^5

so integral becomes sinh y / (sinh y)^5 dy

which is 1/ ( sinh y)^4 or ( cosech y )^4

I have tried to simplify these two integrals, one by putting cosh^2 y - sinh ^2 y in the numerator to replace the one in 1/ (sinh y)^4. But couldn't seem to get anywhere with the results. Is there some clever trick to apply to where I got down. I have spent way to long on this question and would like to see how someone else would work it out and give me some insight to there thinking.

Kind regards.

Small mistake - it should be $\int \text{csch}^{4} y \ dy$ though I suspect that was a typo.

Write $\int \text{csch}^{4} y \ dy$ as $\int \text{csch}^{2}y \ \text{csch}^{2}y \ dy$ and now use integration by parts.

Recall that $\frac{d}{dx} \coth x = -\text{csch}^{2} x$.
• Sep 7th 2013, 03:24 PM
tylersmith7690
Re: trig/hyperbolic substitution integral
Is there not a way to keep going only using trigonometric or hyperbolic substitutions. Or is it ok to use integration by parts when the initially question says 'Using a trigonometric or hyperbolic substitution, evaluate the following inde nite integral'.

Kind regards
• Sep 8th 2013, 06:44 PM
tylersmith7690
Re: trig/hyperbolic substitution integral
ok thanks