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Integrate the functions{{{1}-{{{x}}}}}{{{1}+{{{x}}}}}

Question

Integrate the functions $\frac{1 - cos x}{1 + cos x}$

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Integrate the functions

\frac{1 - cos x}{1 + cos x}

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Text solutionVerified

Solution: We know,

cos x = 2 cos^{2} (\frac{x}{2}) - 1 \Rightarrow 1 + cos x = 2 cos^{2} (\frac{x}{2})

Also,

cos x = 1 - 2 sin^{2} (\frac{x}{2}) \Rightarrow 1 - cos x = 2 sin^{2} (\frac{x}{2})

∴ \frac{1 - cos x}{1 + cos x} = \frac{2 sin ^{2} ( \frac{x}{2} )}{2 sin ^{2} ( \frac{x}{2} )} = tan^{2} (\frac{x}{2}) .

∴ \int \frac{1 - cos x}{1 + cos x} d x = \int tan^{2} (\frac{x}{2}) d x

= \int (sec^{2} (\frac{x}{2}) - 1) d x

= \frac{tan ( \frac{x}{2} )}{\frac{1}{2}} - x + c

∴ \int \frac{1 - cos x}{1 + cos x} d x = 2 tan (\frac{x}{2}) - x + c .

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Question Text	Integrate the functions $\frac{1 - cos x}{1 + cos x}$
Answer Type	Text solution:1
Upvotes	150