determine if each formula is right or wrong. give a brief reason for each answer.\na. ∫(6x + 3)^2 dx=(6x +…

Explanation:

Step1: Recall the power - rule for integration

The power - rule for integration is $\int u^n du=\frac{u^{n + 1}}{n+1}+C$ ($n\neq - 1$). If $u = 6x+3$, then $du=6dx$.

Step2: Analyze formula a

For $\int(6x + 3)^2dx$, let $u = 6x+3$, $du = 6dx$, so $\int(6x + 3)^2dx=\frac{1}{6}\int u^2du=\frac{(6x + 3)^3}{18}+C$. The given formula $\int(6x + 3)^2dx=\frac{(6x + 3)^3}{3}+C$ is wrong because $\frac{d}{dx}\left(\frac{(6x + 3)^3}{3}+C\right)=3\times\frac{(6x + 3)^2\times6}{3}=6(6x + 3)^2\neq(6x + 3)^2$.

Step3: Analyze formula b

For $\int3(6x + 3)^2dx$, let $u = 6x+3$, $du = 6dx$. Then $\int3(6x + 3)^2dx=\frac{3}{6}\int u^2du=\frac{(6x + 3)^3}{2}+C$. The given formula $\int3(6x + 3)^2dx=(6x + 3)^3+C$ is wrong because $\frac{d}{dx}((6x + 3)^3+C)=3(6x + 3)^2\times6 = 18(6x + 3)^2\neq3(6x + 3)^2$.

Step4: Analyze formula c

For $\int18(6x + 3)^2dx$, let $u = 6x+3$, $du = 6dx$. Then $\int18(6x + 3)^2dx=\frac{18}{6}\int u^2du=(6x + 3)^3+C$. The formula is correct because $\frac{d}{dx}((6x + 3)^3+C)=3(6x + 3)^2\times6=18(6x + 3)^2$.

Answer:

a. wrong; b. wrong; c. right