7.1 Integration Techniques
Integration by Parts
Question #4 (Medium): Solving Definite Integral
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 band the other function .
Then plugging into the integral of integration by parts is as follows:
∫ ( )[ ( ( )) ( ) ] [ ( ) ( ( ))] ∫ ( ( )) ( ) , where ( ( )) ( ) ,
( ( ) , and ( ), ( ) .
Evaluate the definite integral using integration by parts.
( ) ( )
The integral contains two functions that cannot be simplified any further. However, it is after some kind
of chain rule where ( ) comes from . Trying to relate the two, double angle can be ap