Semidefinite integrals

Besides the definite integral ∫baf(x) dx and the indefinite integral ∫ f(x) dx, there is also a semidefinite integralaf(x) dx. While the definite integral works out to a specific value (as long as f, a, and b are specified), the indefinite and semidefinite integrals still have the variable x in them. While the indefinite integral also depends on an arbitrary C, the definite and semidefinite integrals don't have this. So the semidefinite integral fits in between the other two kinds.

Here is one way to define it:

x=af(x) dx = ∫xt=af(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.


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