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

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Evaluate definite integrals using the chain rule

Full solution

Q. Evaluate

\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).

\newline

Submit Answer

Simplify the integrand: Simplify the integrand. $\newline$ We have the integrand $(x-7)/(x-5)$ . We can perform long division or recognize that this is a proper rational function that can be decomposed into a sum of simpler fractions. We can write $(x-7)/(x-5)$ as $1 - 2/(x-5)$ .
Set up the integral: Set up the integral with the simplified integrand. $\newline$ Now we can write the integral as: $\newline$ $\int_{6}^{12} (1 - \frac{2}{x-5}) \, dx$ $\newline$ This separates into two simpler integrals: $\newline$ $\int_{6}^{12} 1 \, dx - \int_{6}^{12} \frac{2}{x-5} \, dx$
Evaluate first integral: Evaluate the first integral. $\newline$ The integral of $1$ with respect to $x$ from $6$ to $12$ is simply $x$ evaluated from $6$ to $12$ : $\newline$ $\int_{6}^{12} 1 \, dx = [x]_{6}^{12} = 12 - 6 = 6$
Evaluate second integral: Evaluate the second integral. $\newline$ The integral of $\frac{2}{x-5}$ with respect to $x$ from $6$ to $12$ is $2$ times the natural logarithm of the absolute value of $(x-5)$ evaluated from $6$ to $12$ : $\newline$ $\int_{6}^{12} \frac{2}{x-5} \, dx = 2 \times [\ln|x-5|]_{6}^{12}$ $\newline$ $= 2 \times (\ln|12-5| - \ln|6-5|)$ $\newline$ $= 2 \times (\ln|7| - \ln|1|)$ $\newline$ $x$ $0$
Combine results: Combine the results from Step $3$ and Step $4$ . $\newline$ The final answer is the difference between the two results: $\newline$ $6 - 2 \times \ln(7)$
Write final answer: Write the answer in simplest form. $\newline$ Since $\ln(1)$ is $0$ , we can simplify the expression to: $\newline$ $6 - 2 \times \ln(7)$ $\newline$ This is already in simplest form with the logarithm condensed into a single term.

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