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Reptile [31]
3 years ago
8

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

Mathematics
1 answer:
slega [8]3 years ago
8 0

Given:

The expression is:

\dfrac{1-\cos 2x}{1+\cos 2x}

To find:

The integration of the given expression.

Solution:

We need to find the integration of \dfrac{1-\cos 2x}{1+\cos 2x}.

Let us consider,

I=\int \dfrac{1-\cos 2x}{1+\cos 2x}dx

I=\int \dfrac{2\sin^2x}{2\cos^2x}dx         [\because 1+\cos 2x=2\cos^2x,1-\cos 2x=2\sin^2x]

I=\int \dfrac{\sin^2x}{\cos^2x}dx

I=\int \tan^2xdx                      \left[\because \tan \theta =\dfrac{\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 \dfrac{1-\cos 2x}{1+\cos 2x} is I=\tan x-x+C.

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