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Evaluate int_(6)^(12)(x-7)/(x-5)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Evaluate ∫612x−7x−5dx \int_{6}^{12} \frac{x-7}{x-5} d x . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

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Full solution

Q. Evaluate ∫612x−7x−5dx \int_{6}^{12} \frac{x-7}{x-5} d x . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
  1. Integrating Terms Separately: The integral of a constant 11 with respect to xx is simply xx. The integral of −2(x−5)-\frac{2}{(x-5)} is −2-2 times the natural logarithm of the absolute value of (x−5)(x-5). We will integrate each term separately.\newline∫(x−7)(x−5)dx=∫1dx−2∫1(x−5)dx\int\frac{(x-7)}{(x-5)} dx = \int 1 dx - 2\int\frac{1}{(x-5)} dx\newline=x−2ln⁡∣x−5∣+C= x - 2\ln|x-5| + C
  2. Evaluating Definite Integral: Now we need to evaluate the definite integral from 66 to 1212. \newline∫612x−7x−5 dx=[x−2ln⁡∣x−5∣] from 6 to 12\int_{6}^{12} \frac{x-7}{x-5} \, dx = [x - 2\ln|x-5|] \text{ from } 6 \text{ to } 12\newline= [12−2ln⁡∣12−5∣]−[6−2ln⁡∣6−5∣][12 - 2\ln|12-5|] - [6 - 2\ln|6-5|]\newline= 12−2ln⁡(7)−6+2ln⁡(1)12 - 2\ln(7) - 6 + 2\ln(1)
  3. Simplifying Expression: We simplify the expression by performing the arithmetic operations and using the fact that ln⁡(1)=0\ln(1) = 0.12−2ln⁡(7)−6+2ln⁡(1)=6−2ln⁡(7)12 - 2\ln(7) - 6 + 2\ln(1) = 6 - 2\ln(7)
  4. Final Answer: Now we can write the final answer in simplest form.\newlineThe integral of x−7x−5\frac{x-7}{x-5} from 66 to 1212 is 6−2ln⁡(7)6 - 2\ln(7).

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