022 Sample Final A, Problem 3

Find the antiderivative: $\int {\frac {6}{x^{2}-x-12}}$

Foundations:
1) What does the denominator factor into? What will be the form of the decomposition?
2) How do you solve for the numerators?
3) What special integral do we have to use?
Answer:
1) Since $x^{2}-x-12=(x-4)(x+3)$  , and each term has multiplicity one, the decomposition will be of the form: ${\frac {A}{x-4}}+{\frac {B}{x+3}}$
2) After writing the equality, ${\frac {6}{x^{2}-x-12}}={\frac {A}{x-4}}+{\frac {B}{x+3}}$, 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 $\int {\frac {c}{x-a}}dx=c\ln(x-a)$ , for any numbers c, a.

Solution:

Step 1:
First, we factor $x^{2}-x-12=(x-4)(x+3)$

Step 2:
Now we want to find the partial fraction expansion for ${\frac {6}{(x-4)(x+3)}}$ , which will have the form ${\frac {A}{x-4}}+{B}{x+3}$
To do this we need to solve the equation $6=A(x+3)+B(x-4)$
Plugging in -3 for x to both sides we find that $6=-7B$   and   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 B = -\frac{6}{7}} .
Now we can find A by plugging in 4 for x to both sides. This yields 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 6 = 7A}  , so 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 A = \frac{6}{7}}
Finally we have the partial fraction expansion: 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{6}{7(x - 4)} - \frac{6}{7(x + 3)}}

Step 3:

Now to finish the problem we integrate each fraction to get: 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{6}{x^2 -x -12} dx = \int \frac{6}{7(x - 4)}dx - \int \frac{6}{7(x + 3)}dx }   to get  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}{7}\ln(x - 4) - \frac{6}{7}\ln(x + 3)}

Step 4:
Now make sure you remember to add the 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 + C} to the integral at the end.

Final Answer:
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}{7}\ln(x - 4) - \frac{6}{7}\ln(x + 3) + C}

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