Page 1: Advanced Integration Techniques
This page covers two main integration techniques: partial fractions and integration by parts, along with the tabular integration method.
Integrating Using Linear Partial Fractions
The section begins by explaining the use of algebraic methods to transform integrands.
Definition: Partial fractions rewrite a rational integrand as a sum of rational functions with linear functions in each denominator.
This method is used for quotients of polynomials when the denominator can be factored into linear and non-repeating factors.
Example: An example is provided for integrating (3x² + 4x + 1) / (x² + 5x + 6) dx, demonstrating the step-by-step process of partial fraction decomposition.
Integration by Parts
This section introduces integration by parts as a method to transform integrals into more easily computable forms.
Highlight: The technique is based on the product rule of differentiation.
The general formula for integration by parts is provided: ∫u dv = uv - ∫v du.
Vocabulary: LIATE - An acronym for choosing 'u' in integration by parts, standing for Logarithm, Inverse Trig, Algebraic, Trigonometric, Exponential.
Several examples are given to illustrate the application of integration by parts, including integrals involving trigonometric and logarithmic functions.
Tabular Integration
The page concludes with an introduction to tabular integration, a method for efficiently performing repeated integration by parts.
Definition: Tabular integration is a technique that integrates repeatedly without several steps, using a table format.
An example is provided to demonstrate the tabular method for integrating x² e^x.
The document emphasizes that these integration techniques can be applied to both definite and indefinite integrals, and may be used with literal functions presented in tabular form.
