Find the integration of (1-cos2x)/(1+cos2x)

Question

asked Nov 13, 2022 153k views

1 Answer

Gilad Novik · Answer 1 · 2022-11-16T15:13:05+0000

Given:

The expression is:

$(1-\cos 2x)/(1+\cos 2x)$

To find:

The integration of the given expression.

Solution:

We need to find the integration of
$(1-\cos 2x)/(1+\cos 2x)$ .

Let us consider,

$I=\int (1-\cos 2x)/(1+\cos 2x)dx$

$I=\int (2\sin^2x)/(2\cos^2x)dx$
$[\because 1+\cos 2x=2\cos^2x,1-\cos 2x=2\sin^2x]$

$I=\int (\sin^2x)/(\cos^2x)dx$

$I=\int \tan^2xdx$
$\left[\because \tan \theta =(\sin \theta)/(\cos \theta)\right]$

It can be written as:

$I=\int (\sec^2x-1)dx$
$[\because 1+\tan^2 \theta =\sec^2 \theta]$

$I=\int \sec^2xdx-\int 1dx$

$I=\tan x-x+C$

Therefore, the integration of
$(1-\cos 2x)/(1+\cos 2x)$ is
$I=\tan x-x+C$ .