You have not done anything wrong. All you need is a little more simplification to get to the desired result.
Taking as a common term,if you simplify the terms further, you will get the value you have stated at the bottom.
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?
You will see that if you type in into Wolfram|Alpha and scroll down to the "alternate forms" section, you will see that there will be the equation in the form that came from Wolfram|Alpha's integration.