# MAT136H1 Lecture Notes - Product Rule

7.1 Integration Techniques

Integration by Parts

Question #1 (Easy): Solving Indefinite Integral

Strategy

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 , .

Sample Question

Evaluate the integral using integration by parts.

Solution

More complex function that simplifies when its derivative is taken should be assigned to . So here,

. Then

. Taking derivative of gives , and taking the anti-

derivative of gives

.

Given the form of integration by parts:

Here:

. 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:

## Document Summary

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 , ( ). More complex function that simplifies when its derivative is taken should be assigned to . Taking derivative of gives , and taking the anti- derivative of gives. Since the second integral has two functions that cannot be simplified, integration by parts needs to be taken one more time. Therefore, by integration by parts the integral is evaluated as: