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

Problem

(∫_^)((2*x−3)2*d(x))

Solution

1. Expand the integrand by squaring the binomial (2*x−3)2

(2*x−3)2=(2*x)2−2*(2*x)*(3)+3

(2*x−3)2=4*x2−12*x+9

2. Rewrite the integral using the expanded polynomial.

(∫_^)((4*x2−12*x+9)*d(x))

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

(∫_^)(4*x2*d(x))=(4*x3)/3

(∫_^)(−12*x*d(x))=(−12*x2)/2=−6*x2

(∫_^)(9*d(x))=9*x

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

(4*x3)/3−6*x2+9*x+C

Final Answer

(∫_^)((2*x−3)2*d(x))=(4*x3)/3−6*x2+9*x+C

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