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MAIN Properties of Definite IntegralsProperties of Definite Integrals
P1. The value of the definite integral of a given function is a real number, depending on its lower and upper limits only, and is independent of the choice of the variable of integration, i.e.
P2. P3. P4. Let P5. Comparison of two integrals If P6. Rules of Integration If (a) (b) Newton – Leibniz formula Comparing the two formulas of the curvilinear trapezoid area, we make the conclusion: if F ( x ) is primitive for the function f ( x ) on a segment [ a, b ] , then This is the famous Newton – Leibniz formula. It is valid for any function f ( x ), which is continuous on a segment [ a , b ] . (i) If then (ii) If Then
Integration by substitution If the function u = g(x) has a continuous derivative on [a, b] and f is continuous on the range of g, then Steps for Integrating by Substitution—Definite Integrals 1. Choose a substitution u = g(x), such as the inner part of a composite function. 2. Compute 3. Re-write the integral in terms of u and du, with the u-limits of integration. 4. Find the resulting integral in terms of u. 5. Evaluate using the u-limits. No need to switch back to x’s! Date: 2015-01-02; view: 1255
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