How do you do partial fraction decomposition?

Factor the denominator, set up the partial fraction template based on factor types (linear, repeated, quadratic), multiply through to clear denominators, equate coefficients or substitute values, and solve the resulting system of equations.

When can you use partial fraction decomposition?

Partial fraction decomposition applies to proper rational expressions where the degree of the numerator is less than the degree of the denominator. If the fraction is improper, perform polynomial long division first.

Partial Fraction Decomposition Calculator – Step-by-Step Solutions

∫partial.calc

∫ PARTIAL FRACTION DECOMPOSITION CALCULATOR

Partial Decomposition
Calculator

Q: What is partial fraction decomposition?

Partial fraction decomposition breaks a complex rational expression into a sum of simpler fractions. It is essential for integration in calculus, inverse Laplace transforms, and solving differential equations.

Decompose rational expressions into partial fractions instantly with step-by-step solutions. Essential for calculus integration.

// partial_fraction_decomposition()

Enter Your Rational Expression

NUMERATOR (polynomial in x)

DENOMINATOR (factored or factorable polynomial)

EXAMPLES:

// Result

// what_is_partial_fraction_decomposition

Partial fraction decomposition is the process of breaking a rational expression (a fraction where both numerator and denominator are polynomials) into a sum of simpler fractions. This technique is indispensable in calculus for integrating rational functions, in differential equations for solving with Laplace transforms, and in discrete mathematics for computing generating functions.

The calculator above handles the most common case: proper rational expressions with distinct linear factors in the denominator. Real-world partial fraction problems span several case types:

// decomposition_cases

Distinct Linear Factors

A/(x+a) + B/(x+b)

Each distinct linear factor (x+a) gets one partial fraction with constant numerator A.

Repeated Linear Factors

A/(x+a) + B/(x+a)²

A factor (x+a)ⁿ generates n partial fractions from power 1 to n.

Irreducible Quadratic

(Ax+B)/(x²+bx+c)

An irreducible quadratic factor gets a linear numerator Ax+B.

Repeated Quadratic

(Ax+B)/(x²+c) + (Cx+D)/(x²+c)²

Repeated irreducible quadratics need one fraction per power.

∫ Remember: Always check that the rational expression is proper (numerator degree < denominator degree) before applying partial fractions. If improper, perform polynomial long division first to get a polynomial plus a proper fraction.

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// frequently_asked_questions

Partial fraction decomposition converts complex rational integrals into sums of simple integrals of the form ∫A/(x+a)dx = A·ln|x+a| + C, which are trivial to evaluate. Without decomposition, integrating expressions like (3x+5)/[(x+1)(x+2)] would require advanced techniques or be impossible by elementary methods.

After setting up the partial fraction template, multiply both sides by the denominator to clear all fractions. You can then either: (1) substitute strategic x-values that zero out individual factors, isolating one constant at a time — called the cover-up method; or (2) expand, collect by powers of x, and equate coefficients to form a linear system.

The built-in calculator handles linear factor denominators. For quadratic denominators that factor into real linear factors (like x²-4 = (x-2)(x+2)), it works fully. For irreducible quadratic factors (like x²+1), the decomposition requires linear numerators — enter these manually using the step-by-step guide above.

Partial DecompositionCalculator