022 Sample Final A, Problem 3

3) What special integral do we have to use?

Foundations:
1) What does the denominator factor into? What will be the form of the decomposition?
2) How do you solve for the numerators?
Answer:
1) Since $x^{2}-x-12=(x-4)(x+3)$  , and each term has multiplicity one, the decomposition will be of the form: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A}{x - 4} + \frac{B}{x + 3}}
2) After writing the equality, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{6}{x^2 -x - 12} = \frac{A}{x - 4} + \frac{B}{x + 3}} , clear the denominators, and evaluate both sides at x = 4, -3, Each evaluation will yield the value of one of the unknowns.
3) We have to remember that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \frac{c}{x - a} dx = c\ln(x - a)}  , for any numbers c, a.