Question

` ∫  {1+tan}/{ x + log  sec  x   dx} `

Answer 1

` Note: "Here, we are considering" log  x  as log_e x `
\[\text{Let I} = \int\frac{1 + \tan x}{x + \log \sec x}dx\]
\[\text{Putting}\ x + \log \sec x = t\]
\[ \Rightarrow 1 + \frac{\sec x \tan x}{\sec x} = \frac{dt}{dx}\]
\[ \Rightarrow \left( 1 + \tan x \right)dx = dt\]
\[ \therefore I = \int\frac{1}{t}dt\]
\[ = \text{log }\left| t \right| + C\]
\[ = \text{log }\left| x + \text{log }\sec x \right| + C \left[ \because t = x + \text{log }\sec x \right]\]