When solving integrals involving rational functions, we need to decompose the expression into simpler terms. This is typically done through a series of methods such as partial fraction decomposition or direct substitution. Below is an overview of the essential techniques used to approach these integrals:

  • Partial fraction decomposition
  • Polynomial long division
  • Trigonometric substitution

Partial Fraction Decomposition is one of the most effective methods for breaking down complex rational functions. The goal is to express the function as a sum of simpler fractions, which are easier to integrate. For example:

Decompose the fraction 1 / (x^2 - 1) into partial fractions:

1 / (x^2 - 1) = 1 / 2 * (1 / (x - 1) - 1 / (x + 1))

Another important method is Polynomial Long Division, which is used when the degree of the numerator is greater than or equal to the degree of the denominator. The division process helps simplify the rational function into a polynomial plus a proper fraction.

Step Action
1 Perform division of the numerator by the denominator
2 Separate the remainder as a fraction
3 Integrate the resulting terms separately