(intsec xmathrm{d}x)

Explanation:

Step1: Multiply by a form of 1

Multiply $\sec x$ by $\frac{\sec x+\tan x}{\sec x + \tan x}$. So we have $\int\sec xdx=\int\frac{\sec x(\sec x+\tan x)}{\sec x+\tan x}dx=\int\frac{\sec^{2}x+\sec x\tan x}{\sec x+\tan x}dx$.

Step2: Use substitution

Let $u = \sec x+\tan x$. Then $du=(\sec x\tan x+\sec^{2}x)dx$.

Step3: Integrate

The integral $\int\frac{\sec^{2}x+\sec x\tan x}{\sec x+\tan x}dx=\int\frac{du}{u}$. According to the integral formula $\int\frac{du}{u}=\ln|u|+C$.

Step4: Substitute back

Substitute $u = \sec x+\tan x$ back, we get $\ln|\sec x+\tan x|+C$.

Answer:

$\ln|\sec x+\tan x|+C$