Question

\[\int e^x \left[ \sec x + \log \left( \sec x + \tan x \right) \right] dx\]

Answer 1

\[\text{ Let I } = \int e^x \left[ \sec x + \text{ log }\left( \sec x + \tan x \right) \right]dx\]

\[\text{ Here, } f(x) = \text{ log }\left( \sec x + \tan x \right) Put e^x f(x) = t\]

\[ \Rightarrow f'(x) = \sec x \]

\[\text{ let e}^x \text{ log }\left( \sec x + \tan x \right) = t\]

\[\text{ Diff  both  sides  w . r . t x }\]

\[ e^x \text{ log }\left( \sec x + \tan x \right) + e^x \frac{1}{\sec x + \tan x}\left( \sec x + \tan x + \sec^2 x \right) = \frac{dt}{dx}\]

\[ \Rightarrow \left[ e^x \log\left( \sec x + \tan x \right) + e^x \left( \sec x \right) \right]dx = dt\]

\[ \Rightarrow e^x \left[ \sec x + \log\left( \sec x + \tan x \right) \right]dx = dt\]

\[ \therefore \int e^x \left[ \sec x + \text{ log} \left( \text{ sec x} + \tan x \right) \right]dx = \int dt\]

\[ = t + C\]

\[ = e^x \text{ log }|\left( \sec x + \tan x \right) + C\]