Integrals — Class 12 Maths Solution

: Let $I = \int {\cfrac{{\cos 2x + 2{{\sin }^2}x}}{{{{\cos }^2}x}}dx}$

$= \int {\cfrac{{\left( {{{\cos }^2}x + {{\sin }^2}x} \right) + 2{{\sin }^2}x}}{{{{\cos }^2}x}}} dx$

$= \int {\cfrac{{{{\cos }^2}x + {{\sin }^2}x}}{{{{\cos }^2}x}}} dx = \int {\cfrac{1}{{{{\cos }^2}x}}dx = \int {{{\sec }^2}xdx} } = \tan x + C$