From the previous two exercises (here and here) we know
Use these results to establish the following formulas:
( Note: There is an error in my edition of the book for part (c). It requests we prove , omitting the in the first term of the result. This is just an error in the book since the formula as stated is false.)
- Using the second formula, and the triple angle identity for cosine ( which implies , derived in this exercise), we have
- Using the second formula above and then the first we have (and also recalling the trig identity which implies ),
- Using the second formula above and then part (a) we have
So what happens to the arbitrary constant after integrating? It looks like it’s assumed to be 0 in all 3 of these problems.