Evaluate the Integral

Problem

(∫_−3^2)(x2+x−6*d(x))

Solution

1. Find the antiderivative of the integrand x2+x−6 using the power rule for integration.

(∫_^)(x2+x−6*d(x))=(x3)/3+(x2)/2−6*x

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

[(x3)/3+(x2)/2−6*x]2(−3)

3. Substitute the upper limit x=2 into the expression.

((2)3)/3+((2)2)/2−6*(2)=8/3+2−12

8/3−10=−22/3

4. Substitute the lower limit x=−3 into the expression.

((−3)3)/3+((−3)2)/2−6*(−3)=−9+9/2+18

9+9/2=27/2

5. Subtract the lower limit value from the upper limit value to find the definite integral.

−22/3−27/2=−44/6−81/6

−125/6

Final Answer

(∫_−3^2)(x2+x−6*d(x))=−125/6

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