Evaluate the Integral integral of (3x-2)^2 with respect to x

Problem

(∫_^)((3*x−2)2*d(x))

Solution

1. Expand the binomial expression (3*x−2)2 using the square of a difference formula (a−b)2=a2−2*a*b+b2

(3*x−2)2=9*x2−12*x+4

2. Rewrite the integral by substituting the expanded polynomial back into the integrand.

(∫_^)((9*x2−12*x+4)*d(x))

3. Apply the sum rule for integration, which allows the integral of a sum to be evaluated as the sum of the integrals.

(∫_^)(9*x2*d(x))−(∫_^)(12*x*d(x))+(∫_^)(4*d(x))

4. Integrate each term using the power rule (∫_^)(xn*d(x))=(x(n+1))/(n+1) and the constant rule.

(9*x3)/3−(12*x2)/2+4*x+C

5. Simplify the coefficients of each term to find the final expression.

3*x3−6*x2+4*x+C

Final Answer

(∫_^)((3*x−2)2*d(x))=3*x3−6*x2+4*x+C

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