The Newton-Leibniz theorem

Theorem: Let function be a continuous function in the closed interval and is an antiderivative of the, that is, then

.

Proof. Partition into parts by the points

,

Write the identify

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Using Lagrange’s theorem to the expression in each brackets we have

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Denote

;

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Find limit of both parts of equality when tends to zero

;

or

Example 10.1 Evaluate integral .

Solution:

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