∫ from 0 to 1 of 1/sqrt(4-x^2) dx
I know from Wolfram Alpha that the answer should be π/6, and that the indefinite integral would be arcsin(x/2).
I see the connection between this and the antiderivative of arcsin, I'm just not sure how to handle the 4 rather than the 1--which technique to use.
∫ 1/sqrt(1-x^2)dx is arcsin x
The Attempt at a Solution
Tried a few ways.
1st attempt was to rewrite as ∫ (4-x^2)^(-1/2)dx then reverse the power and chain rules to get sqrt(4-x^2)/2x from 0 to 1=sqrt(3)/2. Knew this was incorrect but trying to get my thoughts flowing.
2nd attempt was with integration by parts, with u=1/sqrt(4-x^2) and dv=dx...etc. This one ended up pretty ugly though, as vdu was worse than the original equation.
3rd attempt was with u-substitution; this one looked like it was going to work then took a bad turn. Let u=4-x^2, produces ∫ from 4 to 3 1/sqrt(u) --> -sqrt(u) from 4 to 3, then plugging back the 4-x^2 ended up undefined.
I was thinking of partial fractions but the sqrt throws me off. I'm studying for a placement exam and this is a question from a past exam in the class.
Thanks to everyone very much for your help!