Integrate cosech^2x

[aero2146]

Integrate cosech^2x

Thanks for the help!

 

[Ishuda]

Thanks for the help!

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

 

[aero2146]

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

Sorry for not reading "Read Before Posting", I'm new around here.

I'm in the second semester, in first year of university doing Actuarial studies. This was in a calculus component.

The real question that I was trying to do was: Integrate 1/(x^2*sqrt(x^2+16)) [Sorry I don't know Latex]

I've tried doing the question using the substitution x = 4tan(u). This worked however ended up being quite long. So I thought of using x = 4sinh(u), as as substitution but this ends up with me having to integrate cosech^2(u).

I've looked up and know that -coth(u) is the integration of cosech^2(u), I'm not sure how to get there.

Any help is greatly appreciated.

 

[ksdhart]

For the integral of csch²(x), I can think of two methods you can try. First, you could use the integration by parts formula. Recall that you can write the expression like this:

$\displaystyle \int csch^2\left(x\right)dx=\int \:csch\left(x\right)\cdot csch\left(x\right)$

Then can you find appropriate substitutions for u and for dv?

Alternatively, you can rewrite the expression like this:

$\displaystyle \int \:csch^2\left(x\right)dx=\int \:\left(\frac{2}{e^x-e^{-x}}\right)^2dx$

This second method might be helpful if you don't know the integrals of the hyperbolic trig functions.