INTEGRATION BY PARTS Learning Outcomes Using product rule
INTEGRATION BY PARTS Learning Outcomes: • Using product rule for differentiation, know how to create formula for integration by parts • Be able to integrate a product of functions using method of integration by parts.
PRODUCT RULE →Integration by Parts If u and v are functions of x, then Integrating throughout, with respect to x Rearranging, we have:
This formula allows us to turn a complicated integral into more simple ones. We must make sure we choose u and dv carefully. Function u is chosen so that Priorities for Choosing u 1. Let u = ln x 2. Let u = xn 3. Let u = enx is simpler than u.
Example 1: Find We could let u = x or u = sin 2 x. In general, we choose the one that allows to be of a simpler form than u. So for this example, we choose u = x and dv = sin 2 x dx.
Example 1: Find Substituting into the integration by parts formula, we get:
Example 2: Find We could let u = x or u = In general, we choose the one that allows to be of a simpler form than u. So for this example, we choose u = x and dv =
Example 2: Find Substituting into the integration by parts formula, we get:
Example 3: Find Priorities for Choosing u 1. Let u = ln x 2. Let u = xn 3. Let u = enx We could let u = x 2 or u = Recall priorities: We must choose u = ln 4 x and dv = x 2 dx
Example 3: Find Substituting into the integration by parts formula, we get:
Example 4: Find
Example 5: Find
Example 6: Find
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