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Find the value of int_(6)^(8)(3)/(x-4)dx. Write your answer as the logarithm of a single number in simplest form. Answer: ln(◻)

Find the value of ∫683x−4dx \int_{6}^{8} \frac{3}{x-4} d x . Write your answer as the logarithm of a single number in simplest form.\newlineAnswer: ln⁡(□) \ln (\square)

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Q. Find the value of ∫683x−4dx \int_{6}^{8} \frac{3}{x-4} d x . Write your answer as the logarithm of a single number in simplest form.\newlineAnswer: ln⁡(□) \ln (\square)
  1. Integrate Function: Now we integrate the function 1x−4\frac{1}{x-4} with respect to xx.\newlineThe antiderivative of 1x−4\frac{1}{x-4} is ln⁡∣x−4∣\ln|x-4|, so the integral from 66 to 88 is:\newline3×[ln⁡∣x−4∣]3 \times [\ln|x-4|] evaluated from 66 to 88
  2. Evaluate Antiderivative: We now evaluate the antiderivative at the upper and lower limits of the integral and subtract the lower evaluation from the upper evaluation.\newline3×[ln⁡∣8−4∣−ln⁡∣6−4∣]3 \times [\ln|8-4| - \ln|6-4|]\newline= 3×[ln⁡∣4∣−ln⁡∣2∣]3 \times [\ln|4| - \ln|2|]\newline= 3×[ln⁡(4)−ln⁡(2)]3 \times [\ln(4) - \ln(2)]
  3. Combine Logarithmic Terms: We can use the properties of logarithms to combine the terms. The difference of logarithms is the logarithm of the quotient of the arguments.\newline3×ln⁡(42)3 \times \ln(\frac{4}{2})\newline= 3×ln⁡(2)3 \times \ln(2)
  4. Apply Power Rule: Since we have a constant multiple of a logarithm, we can use the power rule of logarithms to move the constant inside the logarithm as an exponent.\newlineln⁡(23)\ln(2^3)\newline= ln⁡(8)\ln(8)

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