Integration Practice

6. Integrating Rational Functions

Decomposition and Substitution

This section demonstrates how to integrate rational functions by decomposing the integrand and using substitution and trigonometric identities.

• Key Point 1: The integral can be split into two parts:

  • (A)

  • (B)

• Key Point 2: For (A), use substitution , :

• Key Point 3: For (B), recognize the form :

• Key Point 4: Combine both results:

• Example: Integrate using the above steps.

7. Integrating Trigonometric Functions

Using Trigonometric Identities

Trigonometric integrals often require identities or substitutions for simplification.

• Key Point 1: can be rewritten using .

• Key Point 2:

• Example: Find .

8. Integration by Parts

Product of Polynomial and Exponential

Integration by parts is useful for products of functions, such as polynomials and exponentials.

• Key Point 1: For , let , .

• Key Point 2: Then , .

• Key Point 3: Apply integration by parts:

• Example: Integrate .

9. Integrating Powers of Sine Functions

Reduction Formula and Substitution

Integrals involving powers of sine often use reduction formulas and substitution.

• Key Point 1: For , let , .

• Key Point 2:

• Key Point 3: Substitute , to simplify further.

• Key Point 4: The result is

• Example: Integrate using substitution and reduction.

Additional info: The notes demonstrate standard techniques in calculus for integrating rational, trigonometric, exponential, and polynomial functions, including substitution, trigonometric identities, and integration by parts. These are foundational skills for college-level calculus students.