Evaluate the Integral

Problem

(∫_0^1)((u+2)*(u−3)*d(u))

Solution

1. Expand the integrand by multiplying the binomials (u+2) and (u−3)

(u+2)*(u−3)=u2−3*u+2*u−6

u2−u−6

2. Apply the Power Rule for integration, which states (∫_^)(un*d(u))=(u(n+1))/(n+1) to each term.

(∫_^)((u2−u−6)*d(u))=(u3)/3−(u2)/2−6*u

3. Evaluate the definite integral at the upper limit 1 and the lower limit 0

[(u3)/3−(u2)/2−6*u]10

4. Substitute the limits into the expression.

(1/3−1/2−6*(1))−(0/3−0/2−6*(0))

1/3−1/2−6

5. Simplify the fractions using a common denominator of 6

2/6−3/6−36/6

−37/6

Final Answer

(∫_0^1)((u+2)*(u−3)*d(u))=−37/6

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