Evaluate the Integral

Problem

(∫_0^2)(x*(x−3)*d(x))

Solution

1. Expand the integrand by distributing x into the parentheses.

(∫_0^2)(x2−3*x*d(x))

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

[(x3)/3−(3*x2)/2]20

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

(2/3−(3*(2)2)/2)

4. Substitute the lower limit of integration x=0 into the expression.

(0/3−(3*(0)2)/2)=0

5. Simplify the resulting values to find the final numerical answer.

8/3−12/2

8/3−6

8/3−18/3

−10/3

Final Answer

(∫_0^2)(x*(x−3)*d(x))=−10/3

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