Evaluate the Integral

Problem

(∫_2^4)(x^2−6*x+9*d(x))

Solution

1. Identify the integrand as a polynomial function ƒ(x)=x^2−6*x+9

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

3. Determine the antiderivative F(x)=(x^3)/3−(6*x^2)/2+9*x

4. Simplify the expression for the antiderivative to F(x)=(x^3)/3−3*x^2+9*x

5. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit x=4 and the lower limit x=2

6. Calculate the value at the upper limit: F(4)=4^3/3−3*(4^2)+9*(4)=64/3−48+36=64/3−12

7. Calculate the value at the lower limit: F(2)=2^3/3−3*(2^2)+9*(2)=8/3−12+18=8/3+6

8. Subtract the lower limit value from the upper limit value: (64/3−12)−(8/3+6)

9. Simplify the final result: 56/3−18=56/3−54/3=2/3

Final Answer

(∫_2^4)(x^2−6*x+9*d(x))=2/3

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