Evaluate the Integral

Problem

(∫_2^5)(4−2*x*d(x))

Solution

1. Identify the integrand and the limits of integration. The function to integrate is ƒ(x)=4−2*x and the interval is [2,5]

2. Apply the power rule for integration to find the antiderivative. The power rule states that (∫_^)(xn*d(x))=(x(n+1))/(n+1)

3. Find the antiderivative of each term. The antiderivative of 4 is 4*x and the antiderivative of −2*x is −x2

4. Set up the evaluation using the Fundamental Theorem of Calculus.

F(x)=4*x−x2

5. Substitute the upper limit x=5 into the antiderivative.

F(5)=4*(5)−(5)2

F(5)=20−25=−5

6. Substitute the lower limit x=2 into the antiderivative.

F(2)=4*(2)−(2)2

F(2)=8−4=4

7. Subtract the lower limit value from the upper limit value.

−5−4=−9

Final Answer

(∫_2^5)(4−2*x*d(x))=−9

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