What is the circumference of the circle?
2 answers:
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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 :)