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 sum formula (a+b)^2=a^2+2*a*b+b^2

(3*x+2)^2=9*x^2+12*x+4

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

(∫_^)((9*x^2+12*x+4)*d(x))

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

(∫_^)(9*x^2*d(x))=(9*x^3)/3=3*x^3

(∫_^)(12*x*d(x))=(12*x^2)/2=6*x^2

(∫_^)(4*d(x))=4*x

4. Combine the results and add the constant of integration C

3*x^3+6*x^2+4*x+C

Final Answer

(∫_^)((3*x+2)^2*d(x))=3*x^3+6*x^2+4*x+C

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