Question

Solve: $\int \frac{\sec x}{(\sec x-\tan x)}dx$.

Answer 1

$=\int \frac{\sec x}{\sec x-\tan x}dx$

$=\int \frac{\sec x(\sec x+\tan x)}{(\sec x-\tan x)(\sec x+\tan x)}dx$

$=\int \frac{{\sec }^{2}x+\sec x\tan x}{{\sec }^{2}x-{\tan }^{2}x}dx$

$=\int \frac{{\sec }^{2}x+\sec x\tan x}{1}dx$

$=\int {\sec }^{2}xdx+\int \sec x\tan xdx$

$=\tan x+\sec x+c$.