Chapter 12.3, Problem 2CP

(a)

To determine

Whether the equation, ${\displaystyle \int \frac{3x^{2}dx}{x^3+4}}=\ln |x^3+4|+C$ is true or false.

(b)

To determine

Whether the equation, ${\displaystyle \int \frac{2xdx}{\sqrt{x^2+1}}}=\ln \left|\sqrt{x^2+1}\right|+C$ is true or false.

(c)

To determine

Whether the equation, ${\displaystyle \int \frac{2}{x}dx}=2\ln \left|x\right|+C$ is true or false.

(d)

To determine

Whether the equation, ${\displaystyle \int \frac{x}{x+1}dx}=x{\displaystyle \int \frac{1}{x+1}dx}=x\ln |x+1|+C$ is true or false.

(e)

To determine

Whether the statement,“ To evaluate ${\displaystyle \int \frac{4x}{4x+1}dx}$, our first step is to divide $4x+1$ into $4x$.” is true or false.