4. Integrate with respect to where and 3. Change the original integral in to an integral in 2. Differentiate both sides so 1. Let 5. Change the answer back to While this method of substitution is a very powerful method for solving a variety of problems, we will find that we sometimes will need to modify the method slightly to address problems, as in the following example. Example 1:Compute the following indefinite integral: : : :
35We refer to this method as integration by parts. The following example illustrates its use. Example 2:Use integration by parts method to compute Solution:We note that our other substitution method is not applicable here. But our integration by parts method will enable us to reduce the integral down to one that we can easily evaluate. Let and then and By substitution, we have We can easily evaluate the integral and have And should we wish to evaluate definite integrals, we need only to apply the Fundamental Theorem to the antiderivative. Lesson Summary 1.We integrated composite functions. 2.We used change of variables to evaluate definite integrals. 3.We used substitution to compute definite integrals. Review Questions Compute the integrals in problems #1–8. 126.96.36.199.188.8.131.52.