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

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

and use the following identity:

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

Thank you!