determine if each formula is right or wrong. give a brief reason for each answer.\na. ∫(5x + 6)^2 dx = ((5x…

Explanation:

Step1: Recall the substitution - integration rule

The general formula for $\int u^n du=\frac{u^{n + 1}}{n+1}+C$ ($n\neq - 1$). Let $u = 5x+6$, then $du=5dx$.

Step2: Analyze formula a

For $\int(5x + 6)^2dx$, if we use substitution $u = 5x+6$ and $du = 5dx$, then $\int(5x + 6)^2dx=\frac{1}{5}\int u^2du=\frac{1}{5}\times\frac{u^{3}}{3}+C=\frac{(5x + 6)^3}{15}+C$. So $\int(5x + 6)^2dx=\frac{(5x + 6)^3}{3}+C$ is wrong.

Step3: Analyze formula b

For $\int3(5x + 6)^2dx$, since $\int(5x + 6)^2dx=\frac{1}{5}\int u^2du$ ($u = 5x+6, du = 5dx$), then $\int3(5x + 6)^2dx=\frac{3}{5}\int u^2du=\frac{3}{5}\times\frac{u^{3}}{3}+C=\frac{(5x + 6)^3}{5}+C$. So $\int3(5x + 6)^2dx=(5x + 6)^3+C$ is wrong.

Step4: Analyze formula c

For $\int15(5x + 6)^2dx$, since $\int(5x + 6)^2dx=\frac{1}{5}\int u^2du$ ($u = 5x+6, du = 5dx$), then $\int15(5x + 6)^2dx = 3\int u^2du=3\times\frac{u^{3}}{3}+C=(5x + 6)^3+C$. So $\int15(5x + 6)^2dx=(5x + 6)^3+C$ is right.

Answer:

a. Wrong. Reason: Missing the factor $\frac{1}{5}$ from substitution. b. Wrong. Reason: Incorrect coefficient from substitution. c. Right. Reason: Correct application of substitution - integration.