Evaluate the Integral

Problem

(∫_^)((x+1)*(3*x−2)*d(x))

Solution

1. Expand the integrand by multiplying the binomials using the FOIL method.

(x+1)*(3*x−2)=3*x2−2*x+3*x−2

2. Combine the like terms to simplify the expression before integrating.

3*x2+x−2

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

(∫_^)(3*x2*d(x))+(∫_^)(x*d(x))−(∫_^)(2*d(x))

4. Compute the antiderivative for each term and add the constant of integration C

(3*x3)/3+(x2)/2−2*x+C

5. Simplify the coefficients to reach the final form.

x3+1/2*x2−2*x+C

Final Answer

(∫_^)((x+1)*(3*x−2)*d(x))=x3+1/2*x2−2*x+C

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