Find the antiderivative: 
| Foundations:
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| 1) What does the denominator factor into? What will be the form of the decomposition?
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| 2) How do you solve for the numerators?
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| 3) What special integral do we have to use?
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| Answer:
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| 1) Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{2}-x-12=(x-4)(x+3)}
, and each term has multiplicity one, the decomposition will be of the form: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {A}{x-4}}+{\frac {B}{x+3}}}
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| 2) After writing the equality, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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.
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| 3) We have to remember that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int {\frac {c}{x-a}}dx=c\ln(x-a)}
, for any numbers c, a.
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Solution:
| Step 1:
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First, we factor 
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| Step 2:
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| Now we want to find the partial fraction expansion for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {6}{(x-4)(x+3)}}}
, which will have the form Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {A}{x-4}}+{B}{x+3}}
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| To do this we need to solve the equation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 6=A(x+3)+B(x-4)}
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| Plugging in -3 for x to both sides we find that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 6=-7B}
and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B=-{\frac {6}{7}}}
.
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| Now we can find A by plugging in 4 for x to both sides. This yields Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 6=7A}
, so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A={\frac {6}{7}}}
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| Finally we have the partial fraction expansion: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {6}{x^{2}-x-12}}={\frac {6}{7(x-4)}}-{\frac {6}{7(x+3)}}}
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| Step 3:
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Now to finish the problem we integrate each fraction to get:  to get Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {6}{7}}\ln(x-4)-{\frac {6}{7}}\ln(x+3)}
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| Step 4:
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| Now make sure you remember to add the Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle +C}
to the integral at the end.
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| Final Answer:
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| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {6}{7}}\ln(x-4)-{\frac {6}{7}}\ln(x+3)+C}
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