Evaluate the Integral

Problem

(∫_−4^2)(x2+2*x−8*d(x))

Solution

1. Identify the integrand and the limits of integration. The function to integrate is ƒ(x)=x2+2*x−8 over the interval [−4,2]

2. Find the antiderivative of the function using the power rule for integration, (∫_^)(xn*d(x))=(x(n+1))/(n+1)

F(x)=(x3)/3+x2−8*x

3. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 2 and the lower limit −4

(∫_−4^2)(x2+2*x−8*d(x))=F(2)−F*(−4)

4. Substitute the upper limit x=2 into the antiderivative.

F(2)=2/3+2−8*(2)

F(2)=8/3+4−16

F(2)=8/3−12=−28/3

5. Substitute the lower limit x=−4 into the antiderivative.

F*(−4)=((−4)3)/3+(−4)2−8*(−4)

F*(−4)=−64/3+16+32

F*(−4)=−64/3+48=80/3

6. Subtract the lower limit value from the upper limit value to find the final result.

Result=−28/3−80/3

Result=−108/3=−36

Final Answer

(∫_−4^2)(x2+2*x−8*d(x))=−36

Want more problems? Check here!