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Partial fractions

Partial fractions is a technique used in calculus to break down a rational function into simpler fractions. The idea is to express the rational function as the sum of simpler fractions, each of which has a simpler denominator.

The first step in partial fractions is to factor the denominator of the rational function into irreducible factors. An irreducible factor is a factor that cannot be factored further. For example, the factor x^2 + 1 is irreducible over the real numbers.

Next, we write the rational function as a sum of simpler fractions, where the denominator of each fraction is one of the irreducible factors. The numerator of each fraction is determined by solving for unknown coefficients.

For example, let's consider the rational function:

R(x) = (x^2 + 1)/(x^3 + x^2)

The denominator x^3 + x^2 factors into x^2(x+1), so we can write:

R(x) = A/x + B/(x+1) + Cx/x^2

To find A, B, and C, we can multiply both sides of the equation by the denominator x^3 + x^2 and then substitute in specific values of x. This will give us a system of linear equations that we can solve for A, B, and C.

Once we have found A, B, and C, we can write the rational function R(x) in terms of simpler fractions:

R(x) = A/x + B/(x+1) + Cx/x^2

This technique is particularly useful in integrating rational functions and evaluating definite integrals.

 partial fractions (x^2-4)/(x^4-x)

partial fractions | (x^2 - 4)/(x^4 - x)

(x^2 - 4)/(x^4 - x) = (-3 x - 1)/(x^2 + x + 1) - 1/(x - 1) + 4/x

(4 - x^2)/(x - x^4)

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

-(3 x)/(x^2 + x + 1) - 1/(x^2 + x + 1) - 1/(x - 1) + 4/x