All Flashcards

What is Partial Fraction Decomposition?

Breaking down a rational function into simpler fractions for easier integration.

Define 'Undetermined Coefficients' in PFD.

A method to find the coefficients of the partial fractions by solving a system of equations.

What is a rational function?

A function that can be expressed as a ratio of two polynomials.

What are linear factors in the context of PFD?

Factors in the denominator of the form (ax + b), where a and b are constants.

What is the general form for decomposing $\frac{P(x)}{(x-a)(x-b)}$ ?

$\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$

What is the integral of $\frac{1}{x+a}$ ?

$\int \frac{1}{x+a} dx = \ln|x+a| + C$

General form of PFD with three linear factors?

$\frac{P(x)}{(x-a)(x-b)(x-c)} = \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}$

How to integrate $\frac{2x+1}{(x-1)(x+2)}$ using PFD?

Decompose: $\frac{A}{x-1} + \frac{B}{x+2}$ . 2. Solve for A and B. 3. Integrate each term separately: $\int \frac{A}{x-1} dx + \int \frac{B}{x+2} dx$ .

Steps to decompose $\frac{3x+2}{x^2-13x+42}$ ?

Factor the denominator: $(x-6)(x-7)$ . 2. Set up the decomposition: $\frac{A}{x-6} + \frac{B}{x-7}$ . 3. Solve for A and B.

What is the first step in using PFD to integrate a rational function?

Determine if the integrand is a rational function with linear, non-repeating factors in the denominator.

How do you solve for the unknown coefficients (A, B, etc.) in PFD?

Multiply both sides by the common denominator, then substitute values of x that eliminate other terms and solve for each variable.

How do you integrate the decomposed fractions in PFD?

Integrate each fraction separately, typically resulting in natural logarithm functions: $\int \frac{A}{x-a} dx = A \ln|x-a| + C$ .