Evaluate the Integral integral of (9x+4)^2 with respect to x

Problem

(∫_^)((9*x+4)^2*d(x))

Solution

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

2. Expand the integrand using the square of a binomial formula (a+b)^2=a^2+2*a*b+b^2

(9*x+4)^2=81*x^2+72*x+16

3. Rewrite the integral as a sum of three separate terms.

(∫_^)((81*x^2+72*x+16)*d(x))

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

(∫_^)(81*x^2*d(x))=(81*x^3)/3=27*x^3

(∫_^)(72*x*d(x))=(72*x^2)/2=36*x^2

(∫_^)(16*d(x))=16*x

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

27*x^3+36*x^2+16*x+C

Final Answer

(∫_^)((9*x+4)^2*d(x))=27*x^3+36*x^2+16*x+C

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