Partial Integration and Substitution Rule

Introduction

Integration is a fundamental concept in calculus, used to find areas, volumes, and solve differential equations. Two important techniques for evaluating integrals are the Substitution Rule and Integration by Parts (also called Partial Integration). These methods allow us to solve more complex integrals by transforming them into simpler forms.

Substitution Rule

The substitution rule is used to simplify integrals by making a substitution that reduces the integral to a basic form. It is especially useful when the integrand contains a function and its derivative.

• General Formula:

• Steps for Substitution:

  1. Substitute and compute .

  2. Rewrite the integral in terms of .

  3. Integrate with respect to .

  4. Substitute back to return to the original variable.

Example:

• Let

• So,

Additional Example:

• Let

• So,

Integration by Parts

Introduction

Integration by parts is a technique based on the product rule for differentiation. It is used to integrate products of functions where substitution is not easily applicable.

Formula and Derivation

• Product Rule for Differentiation:

• Integration by Parts Formula:

Or, more simply:

• Choosing and : Choose to be a function that becomes simpler when differentiated, and to be a function that is easy to integrate.

Steps for Integration by Parts

1. Identify and in the integrand.

2. Compute and .

3. Apply the formula: .

4. Integrate the remaining integral.

Examples

• Example 1:

  • Let

  • Apply the formula:

• Example 2:

  • Let

  • Apply the formula:

• Example 3:

  • Let

  • Apply the formula:

• Repeat integration by parts for and solve for the original integral.

Table: Common Choices for (LIATE Rule)

The LIATE rule helps choose in integration by parts:

Summary

• Substitution is best for integrals involving a function and its derivative.

• Integration by Parts is best for products of functions, especially when one function simplifies upon differentiation.

• Both techniques are essential for solving a wide range of integrals in calculus.

Additional info: The notes also include worked examples and color-coded steps to help visualize the process of substitution and integration by parts. The LIATE rule is a common mnemonic for choosing in integration by parts.