perform the indicated multiplication and then integrate. ∫x²(4x + 7) dx

Explanation:

Step1: Expand the integrand

$x^{2}(4x + 7)=4x^{3}+7x^{2}$

Step2: Integrate term - by - term

$\int(4x^{3}+7x^{2})dx=\int4x^{3}dx+\int7x^{2}dx$ Using the power rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have: $\int4x^{3}dx=4\times\frac{x^{3 + 1}}{3+1}=x^{4}$ and $\int7x^{2}dx=7\times\frac{x^{2+1}}{2 + 1}=\frac{7}{3}x^{3}$

Step3: Combine the results

$\int(4x^{3}+7x^{2})dx=x^{4}+\frac{7}{3}x^{3}+C$

Answer:

$x^{4}+\frac{7}{3}x^{3}+C$