MAT136H1 Lecture : 7.1 Integration by Parts Question #4 (Medium)
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Sometimes integration by parts can be applied right away. But sometimes given expression needs to be simplified first, and grouped in certain way so that one of the components can be expressed in a form of the chain rule. In that case, let the function in the form of the chain rule be and the other function . Then plugging into the integral of integration by parts is as follows: ( ( )), and ( ), ( ) . Evaluate the definite integral using integration by parts. The integral contains two functions that cannot be simplified any further. Trying to relate the two, double angle can be applied: of chain rule where ( ) comes from ( ) since it is already in the derivative chain rule form. Put into the form of integration by parts:
