Partial fractions

Partial fraction decomposition is a mathematical technique used to break down a rational expression (a fraction where the numerator and denominator are polynomials) into a sum of simpler fractions, called partial fractions.

Its primary use is in calculus, specifically for integrating rational functions. Breaking down a complex rational function into simpler partial fractions makes the integration process much easier, as the resulting terms can often be integrated using basic rules (like logarithms or arctangents).

You use partial fraction decomposition when you need to integrate a rational function (a polynomial divided by another polynomial) where the degree of the numerator is less than the degree of the denominator (a proper fraction). If the degree of the numerator is greater than or equal to the denominator (an improper fraction), you first perform polynomial long division.

The general steps are:

1. Check if the fraction is proper. If not, divide the polynomials first.
2. Factor the denominator completely into linear factors (ax+b) and irreducible quadratic factors (ax²+bx+c where b²-4ac < 0).
3. Set up the form of the partial fraction decomposition based on the factors.
4. Solve for the unknown constants (like A, B, C) in the numerator(s) using methods such as equating coefficients or substituting roots of the denominator.

The setup depends on the factors of the denominator:

• Linear Distinct Factors (ax+b): For each distinct linear factor, use a term like  A / (ax+b).
• Repeated Linear Factors ((ax+b)ⁿ): For a factor repeated n times, use a sum of n terms:  A₁ / (ax+b) + A₂ / (ax+b)² + ... + An / (ax+b)ⁿ.
• Irreducible Quadratic Factors (ax²+bx+c): For each distinct irreducible quadratic factor, use a term like  (Ax+B) / (ax²+bx+c).
• Repeated Irreducible Quadratic Factors ((ax²+bx+c)ⁿ): For a factor repeated n times, use a sum of n terms:  (A₁x+B₁) / (ax²+bx+c) + (A₂x+B₂) / (ax²+bx+c)² + ... + (Anx+Bn) / (ax²+bx+c)ⁿ.

After setting up the form, multiply both sides of the equation by the original denominator to clear the denominators. Then you can solve for the constants using two main methods:

• Method of Equating Coefficients: Expand the right side of the equation and group terms by powers of x (x², x, constant). Equate the coefficients of corresponding powers of x on both sides of the equation to get a system of linear equations. Solve this system for the constants.
• Method of Substitution (Heaviside Cover-Up Method): Substitute the roots of the original denominator (the values of x that make the denominator zero) into the equation (after clearing denominators). This can directly give you the values of some constants, especially for distinct linear factors. For irreducible quadratic factors or repeated roots, a combination of methods might be needed.

Once you have decomposed the original rational function into a sum of simpler partial fractions, you integrate each partial fraction separately. The integration of these simpler forms is standard:

• Terms like  A / (ax+b)  integrate to  (A/a) ln|ax+b| + C.
• Terms like  B / (ax+b)ⁿ  (n ≠ 1) integrate using the power rule:  B ∫ (ax+b)-n dx.
• Terms like  (Ax+B) / (ax²+bx+c)  often require splitting into two integrals: one that integrates to a logarithm (related to the derivative of the denominator) and one that integrates to an arctangent.

Sum the results of integrating each partial fraction to get the final integral of the original rational function.

If the degree of the numerator is greater than or equal to the degree of the denominator, you must perform polynomial long division first. This will result in a polynomial plus a proper rational fraction. You then apply partial fraction decomposition only to the proper rational fraction part. The integral will be the integral of the polynomial plus the integral of the partial fractions.