Calculator Inputs
Enter the numerator coefficients from highest degree to constant. Add denominator factors using linear or quadratic factor cards.
Example Data Table
| Numerator |
Denominator factors |
Expected partial form |
Use case |
| 5, 9, 7 |
(x - 1)(x + 2) |
5 + 7/(x - 1) - 3/(x + 2) |
Improper rational function |
| 1, 0, 1 |
(x + 1)(x² + 1) |
A/(x + 1) + (Bx + C)/(x² + 1) |
Mixed linear and quadratic factors |
| 2, 3 |
(x - 2)² |
A/(x - 2) + B/(x - 2)² |
Repeated linear factor |
Formula Used
The calculator starts with a rational function:
R(x) = P(x) / Q(x)
If the fraction is improper, polynomial division is applied first:
P(x) / Q(x) = S(x) + r(x) / Q(x)
For repeated linear factors, it uses:
A₁/(ax+b) + A₂/(ax+b)² + ... + Aₙ/(ax+b)ⁿ
For repeated quadratic factors, it uses:
(B₁x+C₁)/q(x) + (B₂x+C₂)/q(x)² + ...
Each numerator coefficient is solved through a coefficient identity. Linear terms integrate with logarithms or powers. Quadratic terms integrate with logarithms, arctangent expressions, and reduction formulas.
How to Use This Calculator
- Type numerator coefficients from highest degree to lowest degree.
- Select each denominator factor type.
- Enter a, b, and c values for each factor.
- Use the power field for repeated factors.
- Set graph and definite integral bounds.
- Press the calculate button.
- Review the decomposition, integration, and graph.
- Export results using CSV or PDF buttons.
Partial Fraction Integration Guide
Why Decomposition Matters
Partial fraction integration changes a difficult rational function into smaller terms. Each term has a familiar antiderivative. This is useful in algebra, calculus, engineering, physics, and control work. The method also shows hidden structure. Poles become clear. Repeated factors become clear. Long division is also handled when the numerator degree is too large.
A Clean Algebra Process
The process begins by expanding the denominator factors. The calculator then checks the numerator degree. If needed, it separates a polynomial quotient from a proper rational remainder. The remainder is matched with a full partial fraction template. Unknown constants are found by equating coefficients. This avoids guessing and supports repeated factors.
Linear and Quadratic Factors
Linear factors usually create logarithmic terms. Repeated linear factors can also create negative powers. Quadratic factors use a linear numerator. Their integrals may include logarithmic and arctangent parts. The exact form depends on the discriminant. This makes the tool helpful for mixed denominators and classroom checking.
Graph and Definite Estimate
The graph helps you see vertical breaks, steep regions, and smooth intervals. These features matter before evaluating a definite integral. The calculator checks the denominator near zero during numerical integration. If a singularity appears inside the interval, the estimate is not trusted. This protects users from misleading areas.
Practical Study Value
Students can compare manual steps with automatic results. Teachers can build examples quickly. Analysts can inspect rational models before simplification. The CSV export stores the major algebra results. The PDF button creates a printable summary. Use simple factors first. Then test repeated and quadratic factors for deeper practice.
FAQs
1. What does this calculator decompose?
It decomposes rational functions into partial fractions using denominator factors you enter. It can handle linear factors, repeated linear factors, quadratic factors, and repeated quadratic factors.
2. Why do I enter denominator factors instead of one expression?
Factor input reduces parsing errors. It also helps the calculator build the correct partial fraction template for repeated and quadratic terms.
3. Can it integrate improper rational functions?
Yes. It performs polynomial division first. The quotient is integrated as a polynomial. The remaining proper fraction is decomposed and integrated term by term.
4. What happens with repeated linear factors?
A repeated linear factor creates several terms. For example, a factor squared creates A/(ax+b) and B/(ax+b)². Each term has its own integral.
5. How are quadratic factors handled?
Each quadratic factor receives a linear numerator, such as Bx+C. The integration step can include logarithmic parts, arctangent parts, and reduction formulas for higher powers.
6. Why might a definite integral show a warning?
A warning appears when the interval crosses a singularity or gets too close to one. Rational functions can become unbounded there, so a simple estimate may be invalid.
7. What does the graph show?
The Plotly chart shows the rational function over your selected x range. It skips points near denominator zeros to avoid drawing misleading vertical lines.
8. Can I save my results?
Yes. Use the CSV button for structured data. Use the PDF button for a readable summary containing the main decomposition and integration result.