Integration Practice

6. Integrating Rational Functions

Decomposition and Substitution Methods

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 :

  • Let , so

• Key Point 4: Combine results:

• Example: Integrate using the above steps.

7. Integrating Trigonometric Functions

Using Trigonometric Identities

Trigonometric integrals often require the use of identities to simplify the integrand.

• Key Point 1: can be rewritten using

• Key Point 2:

• Example: Find .

8. Integration by Parts

Product of Polynomial and Exponential Functions

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

• Key Point 1: For , let , .

• Key Point 2: Then , .

• Key Point 3: Apply integration by parts formula:

• Example: Integrate .

9. Integrating Powers of Sine Functions

Reduction Formula and Substitution

Integrals involving powers of sine functions can be solved using substitution and reduction formulas.

• Key Point 1: For , let , .

• Key Point 2:

• Key Point 3: Integrate to get

• Key Point 4: Substitute back :

• Example: Integrate using substitution and reduction.

Additional info: The notes cover standard techniques in Calculus II, including substitution, integration by parts, and trigonometric integrals. The table summarizes the main results and methods for each type of integral.