Partial Fractions Integrator

Math Calculus • Integrals

Written by STEM Calculators Team Published December 28, 2025 Updated February 24, 2026

6. Partial Fractions Integrator

Decomposes rational integrals \(\int \frac{P(x)}{Q(x)}dx\) into partial fractions and integrates term-by-term. Now includes a definite integral mode with bounds \(a,b\) and shaded area on the graph.

Inputs

Numerator \(P(x)\)

Polynomial only: + − * ^, parentheses, variable x, constants pi, e. Implicit multiplication supported: (x+1)(x-2), 2x.

Denominator \(Q(x)\)

Must be a nonzero polynomial. The tool factors \(Q(x)\) numerically (real linear + irreducible quadratics).

Integral mode

Definite mode evaluates \(F(b)-F(a)\) when possible; otherwise uses numerical integration.

Decomposition mode

“Import” is useful if you already decomposed \(P/Q\) elsewhere and want only the integral.

Numeric tolerance

Used for root clustering & coefficient simplification.

Max denom degree

Higher degrees work but may be less stable.

Options

Show steps Plot \(f(x)=P/Q\) Show asymptotes (real poles) Grid Shade \([a,b]\) (definite mode) Try “nice” fractions for coefficients

Presets

Click to auto-fill and evaluate.

Ready

Graph

Drag to pan • wheel/pinch to zoom. In definite mode, shading shows the signed region between \(a\) and \(b\) when the interval does not cross a real pole.

Center \(c\)

Half-width \(w\)

Legend

\(f(x)\) real pole shaded \([a,b]\)

x: 0, y: 0, zoom(px/unit): 80

Result

Enter polynomials \(P(x)\), \(Q(x)\), then click Evaluate.

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6. Partial Fractions Integrator

How to integrate rational functions \(\int \frac{P(x)}{Q(x)}dx\) via long division + partial fractions, including repeated factors and irreducible quadratic factors — plus how definite mode works.

1) Step zero: make the rational function proper

If \(\deg P \ge \deg Q\), do polynomial long division:

\[ \frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}, \qquad \deg R < \deg Q \]

Then:

\[ \int \frac{P}{Q}\,dx = \int S(x)\,dx + \int \frac{R(x)}{Q(x)}\,dx \]

2) Factor the denominator (over the reals)

\[ Q(x) = a \prod_i (x-r_i)^{m_i}\;\prod_j (x^2+p_j x + q_j)^{n_j} \]

Each quadratic \(x^2+p_jx+q_j\) is irreducible over \(\mathbb{R}\) (negative discriminant).

3) Partial fractions template

Linear factor (possibly repeated):

\[ \frac{R(x)}{(x-r)^m} \Rightarrow \frac{A_1}{x-r} + \frac{A_2}{(x-r)^2} + \cdots + \frac{A_m}{(x-r)^m} \]

Irreducible quadratic (possibly repeated):

\[ \frac{R(x)}{(x^2+px+q)^n} \Rightarrow \frac{B_1x+C_1}{x^2+px+q} + \frac{B_2x+C_2}{(x^2+px+q)^2} + \cdots + \frac{B_nx+C_n}{(x^2+px+q)^n} \]

Coefficients are found by clearing denominators and matching coefficients (a linear system).

4) Integrate each standard term

Linear pole:

\[ \int \frac{A}{x-r}\,dx = A\ln|x-r| + C \] \[ \int \frac{A}{(x-r)^k}\,dx = \frac{A}{1-k}(x-r)^{1-k} + C,\quad (k\ne 1) \]

Irreducible quadratic (power 1):

\[ \int \frac{Bx+C}{x^2+px+q}\,dx = \frac{B}{2}\ln(x^2+px+q) + \left(C-\frac{Bp}{2}\right)\frac{2}{\sqrt{4q-p^2}} \arctan\!\left(\frac{2x+p}{\sqrt{4q-p^2}}\right) + C \]

The calculator also supports power \(2\) for irreducible quadratics; higher powers may fall back to numeric definite evaluation.

5) Definite integrals in this tool

If you choose definite mode, the tool computes:

\[ \int_a^b \frac{P(x)}{Q(x)}\,dx = F(b)-F(a) \]

If the interval \([a,b]\) crosses a real pole (vertical asymptote), the integral is improper. This tool warns instead of guessing; use the separate Improper Integral Tester for convergence checks.

6) Example (sample)

\[ \int \frac{x+1}{x^2-x-2}\,dx,\qquad x^2-x-2=(x-2)(x+1) \] \[ \frac{x+1}{(x-2)(x+1)}=\frac{5/3}{x-2}+\frac{2/3}{x+1} \] \[ \int \frac{x+1}{x^2-x-2}\,dx = \frac{5}{3}\ln|x-2|+\frac{2}{3}\ln|x+1|+C \]

Frequently Asked Questions

What does a partial fractions integrator do for P(x)/Q(x)?

It rewrites the rational function as a sum of simpler rational terms based on the factors of Q(x). Each term has a standard integral, so the tool integrates the sum to produce the final result.

What happens if the numerator degree is larger than the denominator degree?

The method starts by making the rational function proper using polynomial long division: P/Q = S(x) + R/Q with deg(R) < deg(Q). The integral becomes integral S(x) dx plus the integral of R/Q.

How does definite mode compute the integral from a to b?

When an antiderivative F(x) is obtained from the partial fractions form, the tool evaluates F(b) - F(a). If the closed-form evaluation is not reliable for the case, it can fall back to numerical integration.

Why might the tool warn about real poles in the interval?

A real pole is a vertical asymptote where Q(x)=0, which can make the definite integral improper if [a,b] crosses that point. In such cases, convergence should be analyzed with an improper-integral workflow.

What does the "Try nice fractions" option change?

It attempts to simplify solved coefficients into cleaner rational-looking values when they are close to simple fractions. This can make the decomposition and the final antiderivative easier to read.