Evaluate the following: dx 1+ - Vidyadip

Question

Evaluate the following:
$\int\frac{\text{dx}}{1+\cos\text{x}}$

✓

Answer

Let $\int\frac{\text{dx}}{1+\cos\text{x}}$
$=\int\frac{\text{x}}{2\cos^2\frac{\text{x}}{2}}$ $\Big[\because\ 1+\cos\text{A}=2\cos^2\frac{\text{A}}{2}\Big]$
$=\frac{1}{2}\int\frac{1}{\cos^2\frac{\text{x}}{2}}\text{dx}$
$=\frac{1}{2}\int\sec^2\frac{\text{x}}{2}\text{dx}$
$=\frac{1}{2}\cdot\tan\frac{\text{x}}{2}\cdot2+\text{C}$ $\big[\int\sec^2\text{x dx}=\tan\text{x}\big]$
$=\tan\frac{\text{x}}{2}+\text{C}$

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