Well new subject in my Calculus 2 class, and I do not understand why I was wrong on my homework. First I'll show you my work and then I'll show the correct answer from wolfram which uses a formula (reduction Formula) I have yet to learn. I only used wolfram to check the answer.

The Start,

Break them up:

Break sin up even further so I can use Trig Identities:

Replace sin squared with a trig identity:

Substitution:

Replace cos(x) with u:

FOIL:

Enter in Newly FOIL'ed equation:

Integrate:

Revert 'u' back to cos(x):

Now in Short from Wolfram Alpha (the correct answer):

Where did I go wrong? How do I do this integral without referring to reduction formula?