Find the average value of the function f(x)=(1)/(x-1) from x=5 to x=9. 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)=(1)/(x-1) from  x=5 to  x=9. Express your answer as a constant times  ln 2. Answer:  ◻ln 2

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Average Value Formula: 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, f(x) = 1/(x-1), a = 5, and b = 9.Calculate Integral: First, we calculate the integral of f(x) = 1/(x-1) from x=5 to x=9. ∫[5 to 9] (1/(x-1)) dx This is a standard integral that can be solved by recognizing it as the integral of 1/u where u = x - 1. Let u = x - 1, then du = dx.Apply u-Substitution: Change the limits of integration to match the u-substitution. When x = 5, u = 5 - 1 = 4. When x = 9, u = 9 - 1 = 8. Now we integrate with respect to u from u=4 to u=8. ∫[4 to 8] (1/u) duEvaluate Integral: The integral of 1/u with respect to u is ln|u|. So we evaluate ln|u| from u=4 to u=8. ln|u| evaluated from 4 to 8 is ln|8| - ln|4|.Simplify Natural Log: We know that ln|8| is ln(2^3) which is 3ln(2), and ln|4| is ln(2^2) which is 2ln(2). So, ln|8| - ln|4| = 3ln(2) - 2ln(2) = ln(2).Multiply by Factor: Now we have the integral result, which is ln(2). We need to multiply this by the factor (1/(b-a)) to find the average value. (1/(9-5)) * ln(2) = (1/4) * ln(2).Final Answer: Simplify the expression to get the final answer. (1/4) * ln(2) = 0.25ln(2).

Find the average value of the function $f(x)=\frac{1}{x-1}$ from $x=5$ to $x=9$ . 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 $f(x)=\frac{1}{x-1}$ from $x=5$ to $x=9$ . Express your answer as a constant times $\ln 2$ . $\newline$ Answer: $\square\ln 2$