Question

Integration of $\frac{\cos x}{\cos 3x}$

Answer 1

$\int \frac{\cos x}{\cos 3x}=I=?$

$\cos 3x=4{\cos }^{3}x-3\cos x$ .. Identity

$\cos 3x=\cos x(4{\cos }^{2}x-3)$

$\therefore I=\int \frac{\cos xdx}{\cos x(4{\cos }^{2}x-3)}$

$=\int \frac{1dx}{4{\cos }^{2}x}-\int \frac{1}{3}dx$

$=\frac{1}{4}\int {\sec }^{2}xdx-\frac{1}{3}\int dx$

$I=\frac{1}{4}\tan x-\frac{1}{3}x+c$

$\therefore \int \frac{\cos x}{cos3x}=\frac{1}{4}\tan x-\frac{1}{3}x+C$