How to integrate $\frac{2x+1}{(x-1)(x+2)}$ using PFD?
1. 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 $\frac{3x+2}{x^2-13x+42}$?
1. 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: $\int \frac{A}{x-a} dx = A \ln|x-a| + C$.
Explain the goal of Partial Fraction Decomposition.
To transform a complex rational function into a sum of simpler fractions that are easier to integrate.
Why is factoring the denominator important in PFD?
Factoring allows us to identify the individual fractions needed for the decomposition.
When should you consider using Partial Fraction Decomposition?
When integrating rational functions where the denominator can be factored into linear, non-repeating factors.
What is the relationship between the degree of the numerator and denominator for PFD?
The degree of the numerator should be less than the degree of the denominator. If not, use long division first.
Explain the purpose of solving for A, B, C, etc. in PFD.
These variables represent the numerators of the decomposed fractions, and finding their values allows us to rewrite the original integral.
What is the role of the '+C' in indefinite integrals?
It represents the constant of integration, accounting for all possible antiderivatives of a function.
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?
$\frac{P(x)}{(x-a)(x-b)(x-c)} = \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}$
