Given the function f(x)=|x-3|, what is the value of int_(2)^(5)f(x)dx ?

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Given the function  f(x)=|x-3|, what is the value of  int_(2)^(5)f(x)dx ?

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Splitting the Integral: To evaluate the definite integral of the function f(x) = |x - 3| from x = 2 to x = 5, we need to consider the behavior of the absolute value function. The function f(x) = |x - 3| changes at x = 3, so we need to split the integral at this point.Determining Function Behavior: We split the integral into two parts: from 2 to 3, and from 3 to 5. For x in [2, 3], the expression inside the absolute value is negative, so f(x) = -(x - 3). For x in [3, 5], the expression inside the absolute value is positive, so f(x) = x - 3.Writing the Integral: We write the integral as the sum of two integrals: ∫ from 2 to 3 of -(x - 3) dx plus ∫ from 3 to 5 of (x - 3) dx.Calculating Integral from 2 to 3: First, we calculate the integral from 2 to 3 of -(x - 3) dx. This is equal to -∫ from 2 to 3 of (x - 3) dx, which is -[(x^2/2 - 3x) evaluated from 2 to 3].Evaluating Antiderivative: Evaluating the antiderivative from 2 to 3, we get -[(3^2/2 - 3*3) - (2^2/2 - 3*2)] = -[(9/2 - 9) - (2 - 6)] = -[(9/2 - 9) - (-4)] = -(-1/2 - 4) = 9/2.

Given the function $f(x)=|x-3|$ , what is the value of $\int_{2}^{5} f(x) d x$ ?\

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Given the function $f(x)=|x-3|$ , what is the value of $\int_{2}^{5} f(x) d x$ ?\