Partial Fraction Integral Calculator

Partial Fraction Decomposition:

\[ \int \frac{dx}{x(x+1)} = \int \left(\frac{A}{x} + \frac{B}{x+1}\right) dx = A \ln|x| + B \ln|x+1| + C \]

Partial Fractions:

Integral Result:

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1. What is Partial Fraction Decomposition?

Partial fraction decomposition is a technique to break down complex rational expressions into simpler fractions that are easier to integrate or differentiate. It's particularly useful for solving integrals of rational functions.

2. How Does the Calculator Work?

The calculator performs partial fraction decomposition on the input expression:

\[ \frac{P(x)}{Q(x)} = \frac{A_1}{(x-a)} + \frac{A_2}{(x-b)} + \cdots + \frac{A_n}{(x-n)} \]

Where:

\( P(x) \) — Numerator polynomial
\( Q(x) \) — Denominator polynomial (factored)
\( A_1, A_2, \ldots, A_n \) — Constants to be determined

Explanation: The method involves expressing the original fraction as a sum of simpler fractions with linear denominators.

3. Importance of Partial Fractions

Details: Partial fractions are essential for integrating rational functions, solving differential equations, and performing inverse Laplace transforms in engineering and physics.

4. Using the Calculator

Tips: Enter the numerator constant and the denominator factors separated by commas. For example, for 1/(x(x+1)), enter numerator "1" and denominator "x,x+1".

5. Frequently Asked Questions (FAQ)

Q1: What types of denominators can be used?
A: This calculator handles distinct linear factors. Repeated roots and quadratic factors require more advanced methods.

Q2: How accurate are the results?
A: Results are mathematically exact for proper inputs with distinct linear factors.

Q3: Can I use variables other than x?
A: Currently the calculator assumes 'x' as the variable in the denominator factors.

Q4: What if my denominator can't be factored?
A: The calculator requires the denominator to be factored into linear components.

Q5: Can this handle improper fractions?
A: No, the numerator degree must be less than the denominator degree for partial fractions.