Calculator Input
Formula Used
For a rational function P(x) / Q(x), the calculator first applies polynomial division when the numerator degree is not smaller than the denominator degree.
P(x) / Q(x) = S(x) + R(x) / Q(x)
The remaining proper fraction is decomposed using real linear and quadratic factor terms.
For repeated linear factors, terms are A1 / L(x), A2 / L(x)^2, and so on.
For repeated quadratic factors, terms are (B1x + C1) / Q2(x), (B2x + C2) / Q2(x)^2, and so on.
The coefficients are solved by matching equal powers after clearing the denominator.
How to Use This Calculator
- Enter numerator coefficients in descending order.
- Enter each denominator factor on a new line.
- Use L,a,b,m for linear factors.
- Use Q,a,b,c,m for quadratic factors.
- Add optional lower and upper bounds for a definite value.
- Press Calculate to view decomposition and integration steps.
- Use CSV or PDF buttons to export the result.
Example Data Table
| Numerator | Factor Lines | Meaning | Expected Setup |
|---|---|---|---|
| 2,3,5 | L,1,-1,1 Q,1,0,1,1 |
(2x^2 + 3x + 5) / ((x - 1)(x^2 + 1)) | A / (x - 1) + (Bx + C) / (x^2 + 1) |
| 1,0,4 | L,1,2,2 | (x^2 + 4) / (x + 2)^2 | Quotient plus A / (x + 2) + B / (x + 2)^2 |
| 3,1 | Q,1,0,4,1 | (3x + 1) / (x^2 + 4) | Logarithm plus arctangent terms |
Article: Partial Fraction Integration
Why This Method Matters
Rational functions appear in algebra, calculus, signals, control work, and many applied models. A rational function is a quotient of two polynomials. Direct integration can be hard when the denominator is factored into several parts. Partial fractions convert the quotient into simpler pieces. Each piece has a known integral form.
What the Calculator Does
This calculator handles improper and proper rational expressions. It begins with polynomial division. The quotient becomes a direct polynomial integral. The smaller remainder is then decomposed over the entered factors. Linear factors receive constant numerators. Quadratic factors receive linear numerators. Repeated factors receive a term for each power.
How Coefficients Are Found
The calculator clears denominators and builds an identity. It then matches coefficients for equal powers of the variable. This creates a linear system. Solving that system gives the unknown constants. This approach is reliable because it uses polynomial equality, not trial substitution alone.
Linear Factor Integration
A term like A / (ax + b) integrates to a logarithm. Higher repeated powers integrate to negative power terms. These rules are direct and fast. They also make repeated roots easier to check.
Quadratic Factor Integration
A quadratic term uses a numerator Bx + C. The calculator splits it into a derivative part and a residual part. The derivative part gives a logarithm. The residual part usually gives an arctangent when the quadratic is irreducible over the real numbers.
Definite Integrals
For definite bounds, the calculator estimates the original rational function numerically. This is useful for repeated quadratics and longer expressions. The interval should not cross a zero of the denominator. Such points create singular behavior and invalid finite areas.
Good Input Practice
Enter factors carefully. The denominator is built from those factors and the scale value. Keep coefficients in standard order. Review the displayed denominator before using final results. Small input errors can change every coefficient in the decomposition.
Learning Value
The result shows quotient, remainder, coefficients, decomposition, and antiderivative. This makes it useful for checking homework, preparing examples, and studying integration patterns step by step.
FAQs
1. What does this calculator integrate?
It integrates rational functions whose denominators are entered as linear or quadratic factors. It also handles repeated factors and polynomial division before decomposition.
2. How do I enter a linear factor?
Use L,a,b,m. This means (ax + b)^m. For example, L,1,-3,2 means (x - 3)^2.
3. How do I enter a quadratic factor?
Use Q,a,b,c,m. This means (ax^2 + bx + c)^m. For example, Q,1,0,4,1 means x^2 + 4.
4. Can it handle improper rational functions?
Yes. It performs polynomial division first. The quotient is integrated as a polynomial, and the remainder is decomposed by partial fractions.
5. Why do I need to enter factors?
Factoring general polynomials is complex. Entering factors gives clearer control, supports repeated factors, and keeps the decomposition method transparent.
6. What happens with repeated factors?
The calculator creates one partial fraction term for each power. For example, (x + 1)^3 creates terms over powers one, two, and three.
7. Are definite integrals exact?
The displayed definite value is a numerical Simpson estimate. It is useful for checking areas, but intervals crossing denominator zeros should be avoided.
8. What exports are available?
You can download the result as CSV or PDF. Both include the rational expression, decomposition, antiderivative, and solved coefficients.