Here is one way to define it:
∫x=a f(x) dx = ∫xt=a f(t) dt.That is, introduce a new variable t and use the old variable x as the upper limit of a definite integal.
One way to state the Fundamental Theorem of Calculus now is to say (for a continuous function on an interval) that semidefinite and indefinite integrals are the same, up to a constant. (Since the semidefinite integral is defined using the definite integral, this statement links the definite and indefinite integrals, as the FTC should.) Specifically, the first part of the theorem states that every semidefinite integral is an indefinite integral, while the second part of the theorem states that every indefinite integral is a semidefinite integral plus a constant.
Semidefinite integrals are handy when solving initial-value problems.
This web page was written in 2015 by Toby Bartels, last edited on 2015 December 2. Toby reserves no legal rights to it.
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