Evaluate the Integral integral of (12x+5)^2 with respect to x

Problem

(∫_^)((12*x+5)2*d(x))

Solution

1. Identify the integral as a power of a linear function, which can be solved using usubstitution.

2. Substitute u=12*x+5 which implies that d(u)=12*d(x) or d(x)=1/12*d(u)

3. Rewrite the integral in terms of u

(∫_^)(u2⋅1/12*d(u))

4. Factor out the constant:

1/12*(∫_^)(u2*d(u))

5. Apply the power rule for integration, (∫_^)(un*d(u))=(u(n+1))/(n+1)

1/12⋅(u3)/3+C

6. Simplify the expression:

(u3)/36+C

7. Back-substitute u=12*x+5 to get the final result in terms of x

((12*x+5)3)/36+C

Final Answer

(∫_^)((12*x+5)2*d(x))=((12*x+5)3)/36+C

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