Question

\[\int e^x \sec x \left( 1 + \tan x \right) dx\]

Answer 1

\[\text{ Let I } = \int e^x \sec x\left( 1 + \tan x \right)dx\]

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

\[\text{ Here, }f(x) = \text{ sec x Put e}^x f(x) = t\]

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

\[\text{ let e}^x \sec x = t\]

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

\[ e^x \sec x + e^x \sec x \tan x = \frac{dt}{dx}\]

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

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

\[ = t + C\]

\[ = e^x \sec x + C\]