Question

` ∫  {sec  x   "cosec " x}/{log  ( tan x) }`  dx

Answer 1

` Note: Here ,  we   are "  considering " log x  as   log_e x` .
      Let I = ` ∫  {sec  x   "cosec " x}/{log  ( tan x) }`  dx
`  "Putting"  "log" \ tan x = t `
\[ \Rightarrow \frac{\sec^2 x}{\tan x} = \frac{dt}{dx}\]
\[ \Rightarrow \text{sec x cosec x dx} = dt\]
\[ \therefore I = \int\frac{1}{t}dt\]
\[ = \text{log }\left| \text{t }\right| + C\]
\[ = \text{log} \left| \text{log} \left( \tan x \right) \right| + C\]