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Use the given theorem to evaluate the definite integral: ∫(9(x² - 4x + 6)) dx from 1 to ?

User Thanya
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Final answer:

To evaluate the given definite integral from 1 to x2, expand and integrate term-by-term, multiplying each term by 9, and then substitute the limits to find the area under the curve.

Step-by-step explanation:

The definite integral of a function represents the area under the curve of the function between two points on the x-axis. To evaluate the integral ∫(9(x² - 4x + 6)) dx from 1 to an unspecified upper limit, we first express the integral in a simplified form. This involves expanding the equation within the integral and then integrating term-by-term.

First, we multiply out the constant: ∫9*(x² - 4x + 6)dx = 9*∫(x² - 4x + 6)dx.

Next, we integrate the function term-by-term from 1 to the unspecified upper limit x2:

  • Integrate x² to get (1/3)x³.
  • Integrate -4x to get -2x².
  • Integrate 6 to get 6x.

After integrating, we multiply each term by 9 to get: 9[(1/3)x³ - 2x² + 6x]. We evaluate this from x = 1 to x = x2, and calculate the final value by substituting the upper and lower limits into the integrated function and finding the difference.

User Noughtmare
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