Algebraic Integrals
Techniques for integrating algebraic expressions
Introduction
Algebraic integrals involve the integration of expressions containing algebraic functions, such as polynomials, rational functions (quotients of polynomials), and expressions with roots. These integrals are fundamental in calculus and have numerous applications in physics, engineering, and other sciences.
On this page, we present techniques and formulas for solving various types of algebraic integrals, from the simplest to the more complex ones.
Polynomial Integrals
Polynomials are the simplest algebraic expressions to integrate. We apply the power rule term by term.
Example
Rational Functions
A rational function is the quotient of two polynomials. To integrate rational functions, we generally use the technique of partial fraction decomposition.
Steps for Partial Fraction Decomposition
- Ensure that the degree of the numerator is less than the degree of the denominator. If not, divide first.
- Factor the denominator into linear and irreducible quadratic factors.
- Write the partial fraction decomposition according to the factors of the denominator.
- Solve for the unknown coefficients.
- Integrate each resulting term.
Common Forms
Example
First, we factor the denominator: x² - 4 = (x-2)(x+2)
Then, we write the partial fraction decomposition:
Solving for A and B, we get A = 4, B = 1
Integrals with Roots
Integrals containing expressions with roots often require specific substitutions to simplify them.
Common Forms
Example
We use the substitution u = 2x+3, so x = (u-3)/2 and dx = du/2
Substituting back u = 2x+3:
Key Formula
Use partial fraction decomposition
For rational functions, partial fraction decomposition is a powerful technique that simplifies integration.