Evaluate the following integrals: 1- 2x 1+ 2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{1-\cos2\text{x}}{1+\cos2\text{x}}\text{dx}$

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Answer

$\int\frac{1-\cos2\text{x}}{1+\cos2\text{x}}\text{dx}$
$=\int\frac{2\sin^2\text{x}}{2\cos^2\text{x}}\text{dx}$
$=\int\tan^2\text{x dx}$
$=\int(\sec^2\text{x}-1)\text{dx}$
$=\int\sec^2\text{x dx}-\int\text{dx}$
$=\tan\text{x}-\text{x}+\text{C}$
$\therefore\ \int\frac{1-\cos2\text{x}}{1+\cos2\text{x}}\text{dx}=\tan\text{x}-\text{x}+\text{C}$

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