Calculator Input
Enter numerator coefficients from highest power to constant. Then activate denominator factor rows. Use one row with a power for repeated factors.
Example Data Table
| Numerator | Factor setup | Partial fraction target | Integral type |
|---|---|---|---|
| 5, 7 | (x - 1)(x + 2) | A/(x - 1) + B/(x + 2) | Natural logarithms |
| 3, 2, 1 | (x + 1)² | A/(x + 1) + B/(x + 1)² | Log and power terms |
| 4, 1 | x² + 1 | (Ax + B)/(x² + 1) | Log and arctangent |
Formula Used
Proper rational function:
R(x) = P(x) / Q(x)
Linear factor rule:
C / (ax + b)^k
Quadratic factor rule:
(Ax + B) / (ax² + bx + c)^k
Linear integral:
∫ C/(ax + b) dx = (C/a) ln|ax + b| + C₀
Repeated linear integral:
∫ C/(ax + b)^k dx = C(ax + b)^(1-k)/(a(1-k)) + C₀
Quadratic split:
Ax + B = α(2ax + b) + β
The calculator first expands the selected denominator factors. It divides the numerator by the denominator when needed. Then it solves a linear system for all unknown partial fraction constants. Finally, it integrates each term using logarithm, power, and arctangent formulas.
How to Use This Calculator
- Type numerator coefficients from highest degree to constant term.
- Activate each denominator factor row needed for your problem.
- Choose linear for
ax + bfactors. - Choose quadratic for
ax² + bx + cfactors. - Use the power field for repeated factors.
- Press the calculate button.
- Read the partial fraction form and final integral.
- Use CSV or PDF buttons to save the result.
Understanding Partial Fractions for Indefinite Integration
Why This Method Matters
Partial fractions make difficult rational integrals easier. A rational function is a quotient of two polynomials. Direct integration may not be clear. The denominator often contains several factors. Each factor can produce a simpler fraction. Those simpler fractions have known antiderivatives. This turns one hard expression into many small expressions. The final answer becomes easier to check. It also shows where logarithms and arctangents come from.
Preparing the Rational Function
The first step is checking degree. If the numerator degree is too high, division is required. The calculator performs that division first. The quotient becomes a polynomial part. The remainder becomes the proper rational part. Proper rational functions are best suited for decomposition. This prevents missing a polynomial term in the integral. It also keeps each later equation balanced.
Working With Denominator Factors
Denominator factors guide the entire setup. A linear factor gives a constant numerator. A repeated linear factor gives several constant terms. A quadratic factor gives a linear numerator. That numerator has two unknowns. Repeated quadratic factors add more paired unknowns. The calculator builds each required term automatically. It then expands every term to compare coefficients. This creates a linear system.
Solving the Constants
Every unknown constant must satisfy the original numerator. The calculator matches coefficients by degree. It uses Gaussian elimination to solve the system. This works well for exact factor structures. Use one factor row with a power for repetition. Do not enter the same factor twice. That may create duplicate unknowns. A duplicate setup can make the system singular.
Integrating Each Term
Linear terms usually create logarithms. Repeated linear terms create negative powers. Quadratic terms are split using the derivative of the denominator. The derivative part creates a logarithm. The remaining constant part creates an arctangent or logarithmic ratio. The sign of the quadratic discriminant decides the form. This calculator shows the combined antiderivative. It also appends the integration constant.
Using the Result Carefully
The answer is an indefinite integral. Many equivalent forms are possible. A different calculator may arrange terms differently. Constants may look different after simplification. That does not always mean the answer is wrong. Different antiderivatives can differ by a constant. For best accuracy, keep enough decimal precision. You can export your work for notes. You can also compare the derivative of the final answer. If differentiation returns the original rational function, the integration is correct.
FAQs
1. What does this calculator do?
It decomposes a rational function into partial fractions. Then it integrates the simpler terms. It supports polynomial division, linear factors, repeated linear factors, and quadratic factor terms.
2. How should I enter the numerator?
Enter coefficients from highest degree to constant term. For example, enter 2, -3, 5 for 2x² - 3x + 5.
3. What is a linear denominator factor?
A linear factor has the form ax + b. Enter values for a and b. The c value is ignored when the row type is linear.
4. What is a quadratic denominator factor?
A quadratic factor has the form ax² + bx + c. The calculator uses a linear numerator above each quadratic factor during decomposition.
5. Can I use repeated factors?
Yes. Use the power field for repeated factors. For example, enter power 3 for (x + 2)³. Do not enter the same factor on many rows.
6. Why does the calculator perform polynomial division?
Partial fractions require a proper rational function. If the numerator degree is greater than or equal to the denominator degree, division creates a polynomial part and a proper remainder.
7. What does the constant C mean?
The final + C is the integration constant. Any indefinite integral needs it because many antiderivatives differ by a constant value.
8. Why do logarithms appear?
Logarithms appear when integrating terms like 1/(ax + b). They also appear from the derivative part of a quadratic denominator.
9. Why do arctangent terms appear?
Arctangent terms appear when the quadratic denominator has a positive completed-square form. This often happens with factors such as x² + 1.
10. Can answers look different from my textbook?
Yes. Antiderivatives can be algebraically equivalent but arranged differently. Log rules, constant absorption, and decimal rounding can change the appearance.
11. How can I check the answer?
Differentiate the final integral. If the derivative simplifies to the original rational function, the integration is correct.
12. Why did I get a singular system error?
The factor setup may contain duplicate factors entered separately. Use one row with a higher power for repeated factors. Also check for invalid zero coefficients.
13. Does the calculator support fractions as inputs?
Yes. You may enter simple values like 1/2 or -3/4. Decimal numbers are also accepted.
14. What can I export?
You can export the calculated rational function, factor setup, quotient, remainder, partial fraction form, and final indefinite integral as CSV or PDF.