2x-1 2x+1 dx= 1. -x+C 2. x+ +C 3. x- +C 4. -x- +C - Vidyadip

Question

$\int\frac{\cos2\text{x}-1}{\cos2\text{x}+1}\text{ dx}=$

$\tan\text{x}-\text{x}+\text{C}$
$\text{x}+\tan\text{x}+\text{C}$
$\text{x}-\tan\text{x}+\text{C}$
$-\text{x}-\cot\text{x}+\text{C}$

✓

Answer

$\text{x}-\tan\text{x}+\text{C}$

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

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