Evaluate the Integral

Problem

(∫_−2^3)((x^2−3)*d(x))

Solution

1. Identify the integrand ƒ(x)=x^2−3 and the limits of integration a=−2 and b=3

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

(∫_^)((x^2−3)*d(x))=(x^3)/3−3*x

3. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit and subtracting the value at the lower limit.

[(x^3)/3−3*x]^3_(−2)

4. Substitute the upper limit x=3 into the expression.

((3)^3)/3−3*(3)=27/3−9=9−9=0

5. Substitute the lower limit x=−2 into the expression.

((−2)^3)/3−3*(−2)=(−8)/3+6=(−8)/3+18/3=10/3

6. Subtract the lower limit result from the upper limit result.

0−10/3=−10/3

Final Answer

(∫_−2^3)((x^2−3)*d(x))=−10/3

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