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 substitution - rule for integration

Let (u = 6x+3), then (du=6dx), and (\int f(6x + 3)dx=\frac{1}{6}\int f(u)du). For (\int(6x + 3)^2dx), using the power - rule for integration (\int u^n du=\frac{u^{n + 1}}{n+1}+C(n\neq - 1)), we have (\int(6x + 3)^2dx=\frac{1}{6}\times\frac{(6x + 3)^3}{3}+C=\frac{(6x + 3)^3}{18}+C).

Step2: Check formula a

Differentiate (\frac{(6x + 3)^3}{3}+C) using the chain - rule. If (y=\frac{(6x + 3)^3}{3}+C), let (u = 6x+3), then (y=\frac{u^3}{3}+C). (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). (\frac{dy}{du}=u^2=(6x + 3)^2) and (\frac{du}{dx}=6), so (\frac{dy}{dx}=6(6x + 3)^2\neq(6x + 3)^2). So formula a is wrong.

Step3: Check formula b

Differentiate ((6x + 3)^3+C) using the chain - rule. Let (u = 6x+3), then (y = u^3+C). (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). (\frac{dy}{du}=3u^2=3(6x + 3)^2) and (\frac{du}{dx}=6), so (\frac{dy}{dx}=18(6x + 3)^2\neq3(6x + 3)^2). So formula b is wrong.

Step4: Check formula c

Differentiate ((6x + 3)^3+C) using the chain - rule. Let (u = 6x+3), then (y = u^3+C). (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). (\frac{dy}{du}=3u^2=3(6x + 3)^2) and (\frac{du}{dx}=6), so (\frac{dy}{dx}=18(6x + 3)^2). So formula c is right.

Answer:

a. Wrong, because (\frac{d}{dx}(\frac{(6x + 3)^3}{3}+C)=6(6x + 3)^2\neq(6x + 3)^2) b. Wrong, because (\frac{d}{dx}((6x + 3)^3+C)=18(6x + 3)^2\neq3(6x + 3)^2) c. Right, because (\frac{d}{dx}((6x + 3)^3+C)=18(6x + 3)^2)