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Question
I need to integrate the following:
sqrt(2-2cost) dt
This looks like a u substitution problem with u = -cost.
Troulble is, we need a sint in the problem since that's the derivatve of cost.
Now if we put a sint in the problem, we must also divide by sint.
That way, we haven't changed the number, just the appearance.
The new problem is
⌠
⌡sqrt(2-2cost) dt =
⌠
⌡sint*sqrt(2-2cost)/sint dt.
After letting u = -cost, du = sint dt. This makes the problem
⌠
⌡sqrt(2-2u)/sqrt(1-uČ) du.
This can be factored into
⌠
⌡sqrt(2(1-u)/((1+u)(1-u))) du and the 1-u in the top and bottom cancels. This leaves
⌠
⌡sqrt(2/(1+u)) du. If we let v = √(1+u), dv = du/(2√(1+u)).
Changing the problem slightly results in
⌠
⌡sqrt(4/(2(1+u))) du =>
⌠
⌡ 2 dv, and the answer is 2v + C.
That's 2√(1+u) + C.
That's 2√(1-cost) + C.
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