1. ## Hyperbolic function help

Here are two problems that I tried but it seems I can not solve them. Thank you a lot for any help!

Problem 1: I need to evaluate the integral
$\displaystyle \int {sinh^4xdx}$

I know I need to use an identity. I tried rewriting it in this form:

$\displaystyle \int {[sinh^2x]^2dx}$

and use the following identity:

$\displaystyle sinh^2x = \frac {1}{2}(cosh2x - 1)$

but I can not seem to reach a solution.

Problem 2: Suppose that A and B are constants. Show that the function x(t)=Acoshkt+Bsinhkt is a solution of the differential equation

$\displaystyle \frac {d^2x}{dt^2}={k^2}x(t)$

Thank you!

2. Originally Posted by hasanbalkan
Here are two problems that I tried but it seems I can not solve them. Thank you a lot for any help!

Problem 1: I need to evaluate the integral
$\displaystyle \int {sinh^4xdx}$

I know I need to use an identity. I tried rewriting it in this form:

$\displaystyle \int {[sinh^2x]^2dx}$

and use the following identity:

$\displaystyle sinh^2x = \frac {1}{2}(cosh2x - 1)$

but I can not seem to reach a solution.
you are correct so far. but this problem requires stamina. keep expanding and you will need to use a similar identity again, but just use it. at the end, you will have a string of terms, each of which you can integrate easily

Problem 2: Suppose that A and B are constants. Show that the function x(t)=Acoshkt+Bsinhkt is a solution of the differential equation

$\displaystyle \frac {d^2x}{dt^2}={k^2}x(t)$

Thank you!
just find x''(t) and plug it into the equation given (plug in x(t) as well). if the equation makes sense (that is, both sides simplify to the same thing), it is a solution

3. I am sorry I could not respond sooner. You were absolutely right. It required a little bit more persistence on my side, and the problem was solved. I got some sleep that same night, and in the morning things looked much better.

Thank you!