Find the integrals of the function 2x + 2 ^2 x ^2 x - Vidyadip

Question

Find the integrals of the function $\frac{{\cos 2x + 2{{\sin }^2}x}}{{{{\cos }^2}x}}$

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Answer

$\int {\frac{{\cos 2x + 2{{\sin }^2}x}}{{{{\cos }^2}x}}dx} $

$= \int {\frac{{\left( {1 - 2{{\sin }^2}x} \right) + 2{{\sin }^2}x}}{{{{\cos }^2}x}}dx} $

using $\cos 2\theta = 1 - 2{\sin ^2}\theta $

$ = \int {\frac{1}{{{{\cos }^2}x}}dx} $

$= \int {{{\sec }^2}xdx} $

= tan x + c

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