Fundamental theorem of calculus (idea)
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Fundamental Theorem of Calculus II
Let f be an integrable function on [a, b]. For x in [a, b], let F(x) be the integral of f(x) from a to x. Then F is continuous on [a, b]. If f is continuous at x0 in (a, b), then F is differentiable at x0 and F'(x0) = f(x0).
Note that FTC I (above w/u) does not imply FTC II and FTC II does not imply FTC I, and so are not equivalent definitions.