MAT136H1 Lecture Notes - Product Rule
7.1 Integration Techniques
Integration by Parts
Question #1 (Easy): Solving Indefinite Integral
Integration by parts is useful when the integral contains two seemingly different function that cannot
further be simplified, so that: .
This is after the form of the Product Rule. Assign simpler variables like and let ,
then by taking derivative of , and by taking the anti-derivative of , .
Evaluate the integral using integration by parts.
More complex function that simplifies when its derivative is taken should be assigned to . So here,
. Taking derivative of gives , and taking the anti-
derivative of gives
Given the form of integration by parts:
. Since the second integral
has two functions that cannot be simplified, integration by parts needs to be taken one more time.
So again let , and
. Then, and
Then simplify the second integral:
Merge this into the first part of the answer:
Therefore, by integration by parts the integral is evaluated as: