Math 2414 Section 7 2 Integration by Parts The Substitution Rule Section 5 5 arises when we reverse the Chain Rule for derivatives In this section we employ a similar strategy and reverse the Product Rule for derivatives The result is an integration technique called integration by parts Consider the indefinite integrals e dx x and xe x dx Integration by Parts Suppose that u and v are differentiable functions Then u dv uv v du Evaluate xe Evaluate x sin x dx x dx When should we use Integration by Parts Loosely when you see two distinct functions and substitution has not worked What should we choose for u There are five types of functions we might consider for u In order of their desirability we have logarithmic functions inverse functions polynomial functions exponential functions and trigonometric functions The mnemonic device LIPET lie pet might help x 2 e x dx Evaluate e sin xdx 2x Evaluate Evaluate x ln x dx Evaluate sin 2 1 x dx sec x dx 3 Evaluate Integration by Parts for Definite Integrals Integration by parts with definite integral presents two options You can use the previous method to find an antiderivative and then evaluate it at the upper and lower limits of integration If you choose this method you must make certain you do not set an antiderivative equal to a definite integral Alternatively the limits of integration can be incorporated directly into the integration by parts process Integration by Parts for Definite Integrals Let u and v be differentiable Then b b b u x v v x u x dx u x v x a x dx a 2 Evaluate ln x dx 1 using both methods a