Find the average value of the function f(x)=(8)/(x-5) from x=1 to x=3. Express your answer as a constant times ln 2. Answer: ◻ln 2

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Find the average value of the function  f(x)=(8)/(x-5) from  x=1 to  x=3. Express your answer as a constant times  ln 2. Answer:  ◻ln 2

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Set Up Integral: To find the average value of a continuous function f(x) on the interval [a, b], we use the formula for the average value of a function on an interval, which is given by: Average value = (1/(b-a)) * ∫[a to b] f(x) dx Here, a = 1, b = 3, and f(x) = (8)/(x-5).Compute Integral: First, we need to set up the integral to calculate the average value: Average value = (1/(3-1)) * ∫[1 to 3] (8)/(x-5) dx This simplifies to: Average value = (1/2) * ∫[1 to 3] (8)/(x-5) dxEvaluate Antiderivative: Now we need to compute the integral of (8)/(x-5) from x=1 to x=3. This is an integral of the form ∫(dx)/(x-c), which is a natural logarithm function. The antiderivative of (1)/(x-c) is ln|x-c|, so the antiderivative of (8)/(x-5) is 8ln|x-5|.Find Average Value: We evaluate the antiderivative from x=1 to x=3: ∫[1 to 3] (8)/(x-5) dx = [8ln|x-5|] from 1 to 3 = 8ln|3-5| - 8ln|1-5| = 8ln|-2| - 8ln|-4| Since ln|-2| = ln(2) and ln|-4| = ln(4), and ln(4) = 2ln(2), we have: = 8ln(2) - 8*2ln(2) = 8ln(2) - 16ln(2) = -8ln(2)Correct Error: Finally, we multiply this result by (1/2) to find the average value: Average value = (1/2) * (-8ln(2)) = -4ln(2) However, we made a mistake in the sign during the evaluation of the integral. The correct evaluation should result in a positive value since we are integrating from a lower to a higher value of x, and the function (8)/(x-5) is negative on this interval. Let's correct this error.

Find the average value of the function $f(x)=\frac{8}{x-5}$ from $x=1$ to $x=3$ . Express your answer as a constant times $\ln 2$ . $\newline$ Answer: $\square\ln 2$

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Find the average value of the function f(x)=8x−5 f(x)=\frac{8}{x-5} f(x)=x−58​ from x=1 x=1 x=1 to x=3 x=3 x=3. Express your answer as a constant times ln⁡2 \ln 2 ln2.\newlineAnswer: □ln⁡2 \square\ln 2 □ln2

Find the average value of the function $f(x)=\frac{8}{x-5}$ from $x=1$ to $x=3$ . Express your answer as a constant times $\ln 2$ . $\newline$ Answer: $\square\ln 2$