6.8 Evaluating Definite Integrals by Substitution

1. 6.8 Evaluating Definite Integrals by Substitution There are two methods to evaluate a definite integral of the form Method 1. First evaluate the indefinite integral by substitution, and then use the relationship To evaluate the definite integral. This procedure does not require any modification of the x-limits of integration.

2. Method 2. Make the substitution u= g(x) and du=g’(x)dx directly in the definite integral, and then Use the relationship u=g(x) to replace the x-limits, x=a and x=b, by corresponding u-limits, u=g(a) and u=g(b). This produces a new definite integral that is expressed entirely in terms Of u.

3. Example: Use the two methods to evaluate Solution by method 1: Let

4. Example: Use the two methods to evaluate Solution by method 2: Let If x = 0, then u = 1 If x = 1, then u = 2 Thus

5. The choice of methods for evaluating definite integrals by substitution is generally A matter of taste, but in the following examples we will use the second method, Since the idea is new.

6. Example: Evaluate Solution: Let u=1+x, du=dx If x=0, u=1 If x=1/4, u=5/4 Thus,

7. Example: Evaluate Solution: Let If x=0, u=1+1=2 If x=ln5, u=1+5=6 Thus,

8. Example: Evaluate Solution: Solution: Let If If Thus,