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Math  /  Calculus

QuestionQuestion
Evaluate ∫02(7f(x)−6g(x))dx\int_{0}^{2}(7 f(x)-6 g(x)) d x given that ∫014f(x)dx=9,∫02f(x)dx=−9,∫014g(x)dx=5\int_{0}^{14} f(x) d x=9, \int_{0}^{2} f(x) d x=-9, \int_{0}^{14} g(x) d x=5, and ∫02g(x)dx=3\int_{0}^{2} g(x) d x=3
Provide your answer below:

Studdy Solution

STEP 1

1. We are given definite integrals of f(x) f(x) and g(x) g(x) over different intervals.
2. We need to evaluate a linear combination of these functions over the interval [0,2][0, 2].

STEP 2

1. Use the properties of definite integrals to separate the given integral.
2. Substitute the known values of the integrals.
3. Calculate the result.

STEP 3

Use the linearity of integrals to separate the given integral:
∫02(7f(x)−6g(x)) dx=∫027f(x) dx−∫026g(x) dx\int_{0}^{2} (7f(x) - 6g(x)) \, dx = \int_{0}^{2} 7f(x) \, dx - \int_{0}^{2} 6g(x) \, dx
Apply the constant multiple rule to move constants outside the integrals:
=7∫02f(x) dx−6∫02g(x) dx= 7 \int_{0}^{2} f(x) \, dx - 6 \int_{0}^{2} g(x) \, dx

STEP 4

Substitute the known values of the integrals:
=7(−9)−6(3)= 7(-9) - 6(3)

STEP 5

Calculate the result:
=−63−18= -63 - 18 =−81= -81
The value of the integral is:
−81\boxed{-81}

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