$\int \sqrt{\sec x - 1} dx$

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$- \log |(C o s x + \frac{1}{2}) + \sqrt{C o s^{2} x + C o s x}| + C$

$\log |(C o s x + \frac{1}{2}) + \sqrt{C o s^{2} x + C o s x}| + C$

$- \log |(C o s x - \frac{1}{2}) + \sqrt{C o s^{2} x + C o s x}| + C$

$\log |(C o s x + \frac{1}{2}) - \sqrt{C o s^{2} x + C o s x}| + C$

answer is A.

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Detailed Solution

$\begin{array}{l} I = \int \sqrt{\sec x - 1} d x = \int \sqrt{\frac{1 - \cos x}{\cos x}} d x \\ = \int \sqrt{\frac{(1 - \cos x)}{\cos x} \times \frac{(1 + \cos x)}{(1 + \cos x)}} d x \\ = \int \sqrt{\frac{1 - \cos^{2} x}{\cos x + \cos^{2} x}} d x \\ = \int \frac{\sin x}{\sqrt{\cos^{2} x + \cos x}} d x \\ Let \cos x = t . Then d (\cos x) = d t or-sinx dx = dt . Therefore, \\ I = \int \frac{- d t}{\sqrt{t^{2} + t}} \\ = - \int \frac{- d t}{\sqrt{{(t + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}}} \\ = - \log ((t + \frac{1}{2}) + \sqrt{{(t + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}} ∣ + C \\ = - \log |(t + \frac{1}{2}) + \sqrt{t^{2} + t}| + C \\ = - \log (\cos x + \frac{1}{2}) + \sqrt{\cos^{2} x + \cos x} ∣ + C \end{array}$