which of the following could be f(x)? choose all possible answers (select all that apply.) ∫(x + 2)dx f(x) =…

Explanation:

Step1: Apply power - rule for integration

The power - rule for integration is $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int(x + 2)dx$, we integrate term by term. $\int xdx+\int 2dx$. Since $\int xdx=\frac{x^{2}}{2}+C_1$ and $\int 2dx=2x + C_2$, then $\int(x + 2)dx=\frac{x^{2}}{2}+2x + C$, where $C = C_1 + C_2$ is an arbitrary constant.

Step2: Check each option

The antiderivative of $x + 2$ is of the form $\frac{x^{2}}{2}+2x + C$. When $C = 0$, $f(x)=\frac{x^{2}}{2}+2x$; when $C = 1$, $f(x)=\frac{x^{2}}{2}+2x + 1$; when $C=-3$, $f(x)=\frac{x^{2}}{2}+2x - 3$.

Answer:

$f(x)=\frac{x^{2}}{2}+2x$, $f(x)=\frac{x^{2}}{2}+2x + 1$, $f(x)=\frac{x^{2}}{2}+2x - 3$