Section 5.5: Substitution Rule

Introduction to the Substitution Rule

The Substitution Rule (also known as the change of variables) is a fundamental technique in integration, allowing the transformation of integrals into simpler forms. It is especially useful when an integral contains a composite function or when the integrand can be rewritten in terms of a new variable.

• Definition: If is differentiable on an interval and is continuous on the range of , then:

• Purpose: To simplify the process of integration by substituting a part of the integrand with a new variable.

Procedure: Substitution Rule (Change of Variables)

1. Identify an inner function within the integrand.

2. Substitute and compute .

3. Rewrite the integral in terms of and .

4. Integrate with respect to .

5. Substitute back the original variable after integrating.

Examples of the Substitution Rule

• Example 1:

  • Let

  • Rewrite:

• Example 2:

  • Let

  • Rewrite:

General Formulas (Integration)

Below are some essential integration formulas commonly used in calculus:

Additional Examples Using Substitution

• Example 3:

  • Let

  • Rewrite:

• Example 4:

  • Standard result:

• Example 5:

  • Standard result:

• Example 6:

  • Standard result:

Summary Table: Common Integrals

Key Points

• The substitution rule is a powerful tool for simplifying integrals, especially when the integrand is a composite function.

• Always remember to substitute back the original variable after integrating with respect to the new variable.

• Familiarity with standard integration formulas is essential for efficient problem-solving in calculus.