Answer:
Yes, you answer is correct! It just needs to be simplified :)
Step-by-step explanation:
So we have the integral:

As you had done, we can split off the numerator:

Using the Pythagorean Identity, this is:

Now, we can do u-substitution. Let u equal cos(x). Thus:

So:

Simplify:

We can then split the terms:

Expand the integral:

Simplify each of the u.
For the left, that is simply u^-1/2.
For the right, it is u^(2-1/2) or u^3/2. Thus:

Reverse Power Rule:

Simplify:

Simplify further:

Distribute the negative:

And substitute back cos(x) for u:

And this is precisely what you got, so well done!
We can simplify this by first multiplying the first term by 5 to get a common denominator. So:

Combine:

Factor out a cos^(1/2)(x) and a 2. Since we factored out a cos^(1/2)(x), we need to subtract their exponents inside. Thus:

Simplify:

Simplify:

And, of course, C:

So:

And we're done :)