Question

Evaluate the following integrals:
$\int \frac{\sec x}{\sec 2x}dx$

Answer 1

$I=\int \frac{\sec x}{\sec 2x}dx$

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

$$\begin{gathered} =\int \frac{1}{\cos x}dx-2\int \frac{\sin x·\sin x}{\cos x}dx \\ =\int \sec x·dx-2\int \tan x·\sec x·dx \\ =\ln \left|\sec x+\tan x\right|-2\sec x+c \end{gathered}$$