find the indefinite integral. check your work by differentiation. ∫3x(7 - x) dx ∫3x(7 - x) dx =

Explanation:

Step1: Expand the integrand

First, expand $3x(7 - x)$ to get $21x-3x^{2}$. So the integral becomes $\int(21x - 3x^{2})dx$.

Step2: Integrate term - by - term

Use the power rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int21x dx$, since $n = 1$, we have $21\times\frac{x^{1+1}}{1 + 1}=\frac{21}{2}x^{2}$. For $\int-3x^{2}dx$, since $n = 2$, we have $-3\times\frac{x^{2+1}}{2+1}=-x^{3}$. So $\int(21x - 3x^{2})dx=\frac{21}{2}x^{2}-x^{3}+C$.

Step3: Check by differentiation

Differentiate $\frac{21}{2}x^{2}-x^{3}+C$ using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$. $\frac{d}{dx}(\frac{21}{2}x^{2}-x^{3}+C)=\frac{21}{2}\times2x-3x^{2}=21x - 3x^{2}$, which is the original integrand.

Answer:

$\frac{21}{2}x^{2}-x^{3}+C$