Integration and Accumulation of Change

Use synthetic division to divide the factor.

Expand the factor using the binomial theorem.

Treat the factor as a repeated linear factor.

Apply the power rule to simplify the factor.

A removable discontinuity

A horizontal asymptote

A vertical asymptote

A slant asymptote

The degree of the numerator is always less than the degree of the denominator.

The degrees of the numerator and denominator are always equal.

The degree of the numerator is always greater than the degree of the denominator.

The degrees of the numerator and denominator can be different.

Perform polynomial long division because of the higher-degree numerator.

Expand the product of linear factors in the denominator.

Substitute sample values for $x$ to simplify integration before decomposition.

Factorize the cubic term in the denominator.

$B=9$

$B=-18$

$B=-9$

$B=18$

$\frac{Ax+B}{(x+2)(x^2+1)}$

$\frac{A}{(x+2)^2} + \frac{B}{x^2+1}$

$\frac{A}{x+2} + \frac{B}{(x^2+1)^2}$

$\frac{A}{x+2} + \frac{Bx+C}{x^2+1}$

Add a constant term to the numerator.

Multiply the numerator and denominator by a common factor.

Factor the denominator into irreducible factors.

Differentiate both the numerator and denominator.

To allow for substitution and easier integration

To ensure uniqueness of the partial fraction decomposition

To determine the degree of the numerator

To simplify the calculations involved in finding the coefficients

Examining if synthetic division between numerator and denominator is possible without remainder.

Determining whether $(x-5)$ is factored out completely in $x^4 -16$.

Factoring $x^4-16$ into $(x^2-4)(x^2+4)$ shows irreducible quadratics.

Checking if $x-5$ is raised to a power greater than one.

It reduces computational complexity by minimizing cubic and higher-degree expressions in denominators post-decomposition.

It aligns with distinct real roots allowing separate terms for each factor upon integration.

It introduces symmetry making it possible to apply trigonometric substitutions more easily after decomposition.

It enables factoring out common terms between numerators during coefficient comparison.