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Partial Fraction Calculator

Decompose rational expressions into partial fractions with full step-by-step solutions. Enter numerator and denominator polynomials — handles distinct roots, repeated roots, and irreducible quadratics.

Distinct linear · Repeated linear · Irreducible quadratic · Polynomial long division · Integration form

Rational Expression

Enter the numerator and denominator as polynomials in x

÷
3x + 5 / (x² − 1)
Quick Examples
Expression Info
Numerator deg—
Denominator deg—
Proper fraction—
Partial fractions—
Partial Fraction Forms
Distinct Linear
1/(x-a)(x-b) → A/(x-a) + B/(x-b)
Repeated Linear
1/(x-a)² → A/(x-a) + B/(x-a)²
Irreducible Quadratic
1/((x-a)(x²+b)) → A/(x-a) + (Bx+C)/(x²+b)
Improper → Long Division First
deg(N) ≥ deg(D): divide first, then decompose remainder
Key Integrals
∫ A/(x−a) dxA·ln|x−a|+C
∫ A/(x−a)² dx−A/(x−a)+C
∫ A/(x−a)ⁿ dxA·(x−a)^(1−n)/(1−n)
∫ A/(x²+a²) dxA/a·arctan(x/a)
∫ Bx/(x²+a²) dxB/2·ln(x²+a²)
Input Format Tips
Use x^2 for powers, * for explicit multiply
Factored denominator: (x-1)*(x+2) or x^2+x-2
Numerator can be 1, a number, or any polynomial in x