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 , then .
P5. Comparison of two integrals
If , then
P6. Rules of Integration
If are continuous function on then
(a) for some constant k.
(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 (Even Function)
then .
(ii) If (Odd Function)
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 . Compute new ulimits of integration g(a) and g(b).
3. Rewrite the integral in terms of u and du, with the ulimits of integration.
4. Find the resulting integral in terms of u.
5. Evaluate using the ulimits. No need to switch back to x’s!
Date: 20150102; view: 1147
