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Integrating Rational Functions Online Calculator

Rational-function integral calculator. Enter numerator/denominator polynomials and interval to integrate numerically, supporting partial fractions.

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∫ P(x)/Q(x) dx
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
Lower limit of integration
Upper limit of integration

📖 Tutorial | Integrating Rational Functions

1. Definition
A rational function is a quotient P(x)/Q(x). First divide polynomials to get a proper fraction, then use partial fraction decomposition into simple terms and integrate each.
2. Symbols
SymbolsMeaning
P(x)Numerator
Q(x)Denominator
Partial fractionsSplit into simple fractions
Proper fractiondeg P < deg Q
3. How it works
  • Adaptive Simpson integrates the rational expression directly;
  • It verifies the result you obtain by partial fractions;
  • Avoid poles inside the interval.
4. Steps
  1. Enter a rational function, e.g. (x+1)/(x^2+1);
  2. Enter [a,b], avoiding denominator zeros;
  3. Click Calculate for the numeric integral.
5. Example
Example: ∫₀¹ (x+1)/(x²+1)dx = ½ln2 + π/4 ≈ 1.1320. Tool: 1.131970.
6. Pitfalls
Never cross a real zero of the denominator (a pole);

❓ FAQ | Integrating Rational Functions

What is partial fraction decomposition?
Split the rational fraction into terms like A/(x−a), B/(x−a)², (Cx+D)/(ax²+bx+c), then integrate each.
Can the tool decompose automatically?
No analytic decomposition; it integrates numerically to compare with yours.
What if the interval hits a denominator zero?
Avoid poles; crossing a pole diverges.
Why split (x+1)/(x²+1)?
x/(x²+1) gives ½ln(x²+1); 1/(x²+1) gives atan x.
Which notation?
Polynomials, fractions, parentheses and common elementary functions.