Browse Questions

# Find the integrals of the functions$\frac{\cos2x+2\sin^2x}{\cos^2x}$

$\begin{array}{1 1}\tan x + c \\\cot x + c \\ \sin x + c \\ \cos x + c \end{array}$

Toolbox:
• $(i)\cos2x=1-2\sin^2 x+c.$
• $(ii)\;\int \sec^2xdx=\tan x+c$.
Given $I=\int\frac{\cos2x+2\sin^2x}{\cos^2x}dx.$

This can be written as ($2\sin^2x=1-\cos 2x$).

$I=\int\frac{\cos2x+(1-\cos 2x)}{\cos^2x}dx.$

On cancelling cos 2x we get,

$\;\;\;=\int\frac{1}{\cos^2x}dx=\int \sec^2xdx.$

On integrating we get,

$\int\frac{\cos 2x+2\sin^2x}{\cos ^2x}dx=\tan x+c.$