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: Analyze formula a

The derivative of (\frac{(6x + 3)^3}{3}+C) using the chain - rule (\frac{d}{dx}(u^n)=nu^{n - 1}\frac{du}{dx}) (where (u = 6x+3) and (n = 3)) is (\frac{d}{dx}(\frac{(6x + 3)^3}{3}+C)=\frac{3(6x + 3)^2\times6}{3}=6(6x + 3)^2\neq(6x + 3)^2). So formula a is wrong.

Step3: Analyze formula b

The derivative of ((6x + 3)^3+C) using the chain - rule is (\frac{d}{dx}((6x + 3)^3+C)=3(6x + 3)^2\times6 = 18(6x + 3)^2\neq3(6x + 3)^2). So formula b is wrong.

Step4: Analyze formula c

The derivative of ((6x + 3)^3+C) using the chain - rule is (\frac{d}{dx}((6x + 3)^3+C)=3(6x + 3)^2\times6=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)