determine the following indefinite integral. check your work by differentiation.\n int(sec^{2}x + 2)dx \n…

Explanation:

Step1: Recall integral rules

We know that $\int\sec^{2}x dx=\tan x + C_1$ and $\int kdx=kx + C_2$ (where $k = 2$ is a constant).

Step2: Apply sum - rule of integration

$\int(\sec^{2}x + 2)dx=\int\sec^{2}x dx+\int 2dx$. Since $\int\sec^{2}x dx=\tan x$ and $\int 2dx=2x$, the result of the integral is $\tan x+2x + C$ (where $C = C_1 + C_2$ is the constant of integration).

Step3: Check by differentiation

The derivative of $\tan x+2x + C$ is $\sec^{2}x+2$ using the derivative rules $\frac{d}{dx}(\tan x)=\sec^{2}x$, $\frac{d}{dx}(2x)=2$ and $\frac{d}{dx}(C)=0$.

Answer:

$\tan x + 2x+C$