#-1/8intcos(2x)dx# can also be taken care of fairly easily, this time with a simple #u# substitution. Note: I will leave out adding the constant, #C# until the end. The last integral, #1/8int(1)dx# can be easily taken care of without any further manipulation.
I will show that at the end if you are interested. Note that these are not the original half-angle identities, but the originals have been manipulated to produce these. I would begin by using half-angle identities: I'll include definitions or explanations of the rules used at the very end in the case that you would find this helpful. It's going to require the use of a few trigonometric identities and rules for integration.
