Integrating using partial fractions

Understanding the concept of partial fractions is a fundamental requirement for this section.
A rational function, which is a ratio of two polynomials, is commonly represented as a partial fraction.
Any rational function can be broken down into simpler fraction components through partial fractions making it easier to work and deal with.
Before you can perform an integration using partial fractions, ensure that the degree of the numerator is less than the degree of the denominator in your rational function.
If the degree of the numerator is greater than or equal to that of the denominator, divide the numerator by the denominator to reduce the degree.
The process to break down the rational function into its partial fractions is known as Decomposition. There are different ways to decompose depending on the denominator.
If the denominator of the rational function consists of distinct linear factors, each factor will form a separate fraction with a constant in the numerator.
If a rational function’s denominator contains repeated linear factors, for each repeated factor, you create distinct partial fractions with different powers of that factor.
For rational functions with quadratic factors in the denominator that cannot be further simplified, you can write the partial fraction as a linear term divided by the quadratic factor.
After you have decomposed the integral into partial fractions, proceed to integrate these fractions individually.
Some integrals of partial fractions can be solved by simple antiderivatives, while others may require techniques like u-substitution, trigonometric substitution, or integration by parts.
Keep in mind that common errors during the partial fraction decomposition stage can dramatically affect the process of integrating these fractions.
Always check to ensure no terms are left out, especially when dealing with repeated or quadratic factors in the denominator, and make sure to keep track of all signs correctly.
If required, use the method of simultaneous equations to solve for unknown constants that may appear in the decomposition process.
Aim to practice regular application of partial fractions in integration as this method can sometimes appear complex, but becomes simpler with regular practice.
Constantly revise the formulas and conversion methods for the different partial fraction scenarios that may appear during the integration process. Be sure to familiarize yourself with the various cases and patterns observed in the partial fractions.