Integrate rational functions step by step using polynomial division, denominator factorization, partial fractions, verification and integration.
Enter the numerator \(P(x)\) and denominator \(Q(x)\) of a rational function \(\frac{P(x)}{Q(x)}\). The calculator will show the algebraic steps.
Partial fraction integration is a method for integrating rational functions. A rational function is a quotient of two polynomials:
When the denominator can be factored, the fraction can often be rewritten as a sum of simpler fractions. These simpler expressions are then easier to integrate.
For example:
Each resulting term can then be integrated using familiar formulas such as the logarithm rule.
The numerator and denominator are interpreted as polynomials in \(x\).
If the numerator has degree greater than or equal to the denominator, polynomial division is performed before partial fractions.
The calculator looks for linear factors and supported quadratic factors.
A coefficient system is constructed and solved algebraically.
The calculator recombines the partial fractions and compares the resulting polynomial with the original numerator.
Linear factors produce logarithmic or power integrals, while supported quadratic factors use the corresponding quadratic integration formulas.
The form of the decomposition depends on how the denominator factors.
More generally, a repeated factor requires one partial-fraction term for each power of that factor.
The numerator over an irreducible quadratic is linear because its degree must be smaller than the degree of the quadratic denominator.
The denominator has two distinct linear factors, so the fraction can be separated into two simple logarithmic terms.
The repeated factor \((x+1)^2\) requires terms involving both \(x+1\) and \((x+1)^2\).
The quadratic \(x^2+1\) is irreducible over the real numbers and uses a linear numerator in its partial-fraction term.
Because the numerator and denominator have the same degree, polynomial long division is performed first.
Partial fractions are normally applied to a proper rational function, where \(\deg P < \deg Q\).
If \(\deg P \geq \deg Q\), polynomial long division separates the expression into a polynomial plus a proper rational fraction.
The polynomial \(S(x)\) can be integrated directly, while the proper fraction \(R(x)/Q(x)\) can then be decomposed using partial fractions.
2, 3.5 and -4x2xx^2 and (x+1)^3(x+1)(x+2)Partial fractions are especially useful when the integrand is a quotient of polynomials.
Other functions may be better handled with different methods:
Partial fraction integration rewrites a rational function as a sum of simpler rational expressions. Each simpler expression can then be integrated using standard formulas.
Partial fractions are commonly used when integrating a rational function, meaning a quotient of two polynomials. Factoring the denominator determines the form of the decomposition.
The calculator performs polynomial long division first. This separates the expression into a polynomial and a proper rational fraction.
Yes. The solution can show polynomial division, denominator factorization, partial-fraction decomposition, algebraic verification and integration.
Yes. For example, denominators containing \((x+1)^2\) can be decomposed into terms involving \(x+1\) and \((x+1)^2\).
Yes. Supported irreducible quadratic factors use a linear numerator in the partial-fraction decomposition. Some repeated-root quadratic cases require additional simplification and may not be fully evaluated by this version of the calculator.
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