Evaluate the Integral

Problem

(∫_−1^2)((x−8*|x|)*d(x))

Solution

1. Split the integral based on the definition of the absolute value function |x| which changes behavior at x=0

2. Define the intervals for the piecewise function where |x|=−x for x<0 and |x|=x for x≥0

3. Rewrite the expression for the first interval [−1,0] as x−8*(−x)=9*x

4. Rewrite the expression for the second interval [0,2] as x−8*(x)=−7*x

5. Set up the sum of the two definite integrals.

(∫_−1^0)(9*x*d(x))+(∫_0^2)(−7*x*d(x))

6. Find the antiderivatives using the power rule.

[(9*x2)/2]0(−1)+[(−7*x2)/2]20

7. Evaluate at the boundaries by subtracting the lower limit value from the upper limit value.

((9*(0)2)/2−(9*(−1)2)/2)+((−7*(2)2)/2−(−7*(0)2)/2)

8. Simplify the arithmetic to find the final value.

(0−9/2)+(−14−0)

9. Combine the terms using a common denominator.

−9/2−28/2=−37/2

Final Answer

(∫_−1^2)((x−8*|x|)*d(x))=−37/2

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