Question

\[\int\frac{\tan x}{\sec x + \tan x} dx\]

Answer 1

\[\int\frac{\tan x}{\sec x + \tan x}dx\]
\[ = \int\frac{\tan x}{\left( \sec x + \tan x \right)} \times \left( \frac{\sec x - \tan x}{\sec x - \tan x} \right)dx\]
\[ = \int\frac{\tan x \left( \sec x - \tan x \right)}{\left( \sec^2 x - \tan^2 x \right)}dx\]
\[ = \int\left( \frac{\sec x \tan x - \tan^2 x}{1} \right)dx\]
\[ = \int\text{sec x }\text{tan  x  dx} - \int\left( se c^2 x - 1 \right)dx\]
\[ = \sec x - \tan x + x + C\]