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

Given the function f(x)=∣x−3∣ f(x)=|x-3| , what is the value of ∫25f(x)dx \int_{2}^{5} f(x) d x ?\

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Find indefinite integrals using the substitution and by partsright chevron icon

Full solution

Q. Given the function f(x)=∣x−3∣ f(x)=|x-3| , what is the value of ∫25f(x)dx \int_{2}^{5} f(x) d x ?\
  1. Splitting the Integral: To evaluate the definite integral of the function f(x)=∣x−3∣f(x) = |x - 3| from x=2x = 2 to x=5x = 5, we need to consider the behavior of the absolute value function. The function f(x)=∣x−3∣f(x) = |x - 3| changes at x=3x = 3, so we need to split the integral at this point.
  2. Determining Function Behavior: We split the integral into two parts: from 22 to 33, and from 33 to 55. For xx in [2,3][2, 3], the expression inside the absolute value is negative, so f(x)=−(x−3)f(x) = -(x - 3). For xx in [3,5][3, 5], the expression inside the absolute value is positive, so f(x)=x−3f(x) = x - 3.
  3. Writing the Integral: We write the integral as the sum of two integrals: ∫23−(x−3) dx\int_{2}^{3} -(x - 3) \, dx plus ∫35(x−3) dx\int_{3}^{5} (x - 3) \, dx.
  4. Calculating Integral from 22 to 33: First, we calculate the integral from 22 to 33 of −(x−3) dx-(x - 3) \, dx. This is equal to −∫23(x−3) dx-\int_{2}^{3} (x - 3) \, dx, which is −[x22−3x]-\left[\frac{x^2}{2} - 3x\right] evaluated from 22 to 33.
  5. Evaluating Antiderivative: Evaluating the antiderivative from 22 to 33, we get −[(32/2−3×3)−(22/2−3×2)]=−[(9/2−9)−(2−6)]=−[(9/2−9)−(−4)]=−(−1/2−4)=9/2-[(3^2/2 - 3\times3) - (2^2/2 - 3\times2)] = -[(9/2 - 9) - (2 - 6)] = -[(9/2 - 9) - (-4)] = -(-1/2 - 4) = 9/2.

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