Chapter Integration of substitution and integration by parts of the definite integral

As we all known, the integration of substitution ( or “change of variables”) and integration by parts are very important tools to evaluate the indefinite integrals. In this section we extend these methods to the definite integrals. 1. Integration of substitution At first, the substitution technique extends to definite integrals

Theorem 1 Theorem 1 (Substitution in a definite integral) Substitution formula of definite integrals

Proof

Notice One important is the necessary change in the limits of integration. The following examples will apply this technique. Example 1 Solution

Example 2 Solution Additivity over intervals

Change of variables Newton-Leibniz formula Example 3 Solution

Example 4 Solution

Example 5

Solution

2. Important simplification formulas Theorem 2

Solution

Notice The properties of definite integrals for odd and even functions provide a easy way to evaluate their definite integrals. The following examples will apply the property. Example 6 Solution even function

odd function Area of unit circle Example 7

Solution odd function Example 8

Solution

Thus, we choice ( D ) Theorem 3 Proof

f (x+T)=f (x) The integral properties of periodic functions provide also a easy way to evaluate some definite integrals. Next example will apply the property. Example 9

Solution

Proof （ 1 ） Let Example 10

3 Integration by parts Theorem 4 Next, the integration by parts extends to definite integrals Integration by parts for definite integrals

Proof This completes the proof

Solution Example 11

Example 12 Solution

Example 13 Solution

Example 14 Solution A

Thus, we choose (A)(A)

Example 15 Solution

Example 16 Solution