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The basic formula of integral calculus

This formula is called the basic formula of integral calculus, or the Newton-Leibniz formula.

The Newton-Leibniz formula gives ample opportunities calculate definite integrals. It is necessary to compute the indefinite integral and then find the difference in the values of the primitive functions. Consider the examples of the calculation of definite integrals.

Example 1.

Example 2.

The basic rules of integration Replacing a variable in the definite integral

THEOREM. Let: 1) a continuous function on [a, b]; 2) the function φ (t) is differentiable on [α, β], where φ '(t) is continuous on [α, β] and the set of values of the function φ (t) is the interval [a, b], 3) φ (α) = a, φ (β) = b. Then the formula

Formula (7.12) called the formula replacing of the variable or substitution of the variable in a definite integral.

Calculate definite integrals by substitution.

Example 3.

DECISION. Substitute t = 1 + х2. Then dt = 2х dx, t = 1 for х = 0 and t = 2 for x = 1. Since the function х = is continuous on [1, 2], and the new integrand is also continuous, and hence into force for it Theorem there is a primitive on this segment. Get

Integration by parts in a definite integral

Theorem. Let u (x) and v (x) have continuous derivatives on [a, b]; Then the formula

Equality (7.13) is a formula for integration by parts in a definite integral. Consider some examples of calculating definite integrals by integration by parts.

Example 4.

DECISION. Put here и = х, v = e-x, тогда dv = -e-xdx и

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