Evalute the following integrals: 1- 1+ dx - Vidyadip

Question

Evalute the following integrals:
$\int\sqrt{\frac{1-\cos\text{x}}{1+\cos\text{x}}}\text{dx}$

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Answer

$\int\sqrt{\frac{1-\cos\text{x}}{1+\cos\text{x}}}\text{dx}$
$=\int\sqrt{\frac{2\sin^2\frac{\text{x}}{2}}{2\cos^2\frac{\text{x}}{2}}}\text{dx}$
$\Big[\because 1-\cos\text{x}=2\sin^2\frac{\text{x}}{2} \ \&\ 1+\cos\text{x}=2\cos^2\frac{\text{x}}{2}\Big]$
$=\int\tan\frac{\text{x}}{2}\text{dx}$
$=-2\text{ln}\Big|\cos\frac{\text{x}}{2}\Big|+\text{C}$

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