Help with a trig integral

OK I'm studying for a calculus exam and I ran across a problem where I thought I did everything right, but I can't know for sure because it's an even problem and I don't have an answer.

__Problem:__

__What I did:__

**Splitting into squares:**

Using cos^2 + sin^2 = 1:

**Substituting:**

**Foiling:**

**Integrating:**

**Unsubstituting:**

__Answer:__

I typed the problem into Wolfram Alpha and it gave me this:

This looks alot different than my answer, and they did it using something called a 'reduction formula', which seems like something I would never remember. Is my answer wrong, or did they just do it differently? Or is Wolfram Alpha not the best place to be checking my answers?

Thanks