To transform a complex rational function into a sum of simpler fractions that are easier to integrate.

Factoring allows us to identify the individual fractions needed for the decomposition.

When integrating rational functions where the denominator can be factored into linear, non-repeating factors.

The degree of the numerator should be less than the degree of the denominator. If not, use long division first.

These variables represent the numerators of the decomposed fractions, and finding their values allows us to rewrite the original integral.

It represents the constant of integration, accounting for all possible antiderivatives of a function.

Breaking down a rational function into simpler fractions for easier integration.

A method to find the coefficients of the partial fractions by solving a system of equations.

A function that can be expressed as a ratio of two polynomials.

Factors in the denominator of the form (ax + b), where a and b are constants.

1. Decompose: $\frac{A}{x-1} + \frac{B}{x+2}$. 2. Solve for A and B. 3. Integrate each term separately: $\int \frac{A}{x-1} dx + \int \frac{B}{x+2} dx$.

1. Factor the denominator: $(x-6)(x-7)$. 2. Set up the decomposition: $\frac{A}{x-6} + \frac{B}{x-7}$. 3. Solve for A and B.

Determine if the integrand is a rational function with linear, non-repeating factors in the denominator.

Multiply both sides by the common denominator, then substitute values of x that eliminate other terms and solve for each variable.

Integrate each fraction separately, typically resulting in natural logarithm functions: $\int \frac{A}{x-a} dx = A \ln|x-a| + C$.