$\int \frac{d x}{cos x - sin x}$ is equal to

\frac{1}{\sqrt{2}} log ∣ ∣ tan (\frac{x}{2} - \frac{3 π}{8}) ∣ ∣ + c

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\frac{1}{\sqrt{2}} log ∣ ∣ cot (\frac{x}{2}) ∣ ∣ + c

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\frac{1}{\sqrt{2}} log ∣ ∣ cot (\frac{x}{2} - \frac{3 π}{8}) ∣ ∣ + c

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\frac{1}{\sqrt{2}} log ∣ ∣ tan (\frac{x}{2} + \frac{3 π}{8}) ∣ ∣ + c

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Solution

The correct option is A $\frac{1}{\sqrt{2}} log ∣ ∣ tan (\frac{x}{2} - \frac{3 π}{8}) ∣ ∣ + c$
$cos x - sin x = \sqrt{2} (\frac{1}{\sqrt{2}} cos x - \frac{1}{\sqrt{2}} sin x) = \sqrt{2} cos (\frac{π}{4} - x)$
$\int \frac{d x}{cos x - sin x} = \frac{1}{\sqrt{2}} \int sec (\frac{π}{4} - x) d x = \frac{1}{\sqrt{2}} log ∣ sec (\frac{π}{4} - x) + tan (\frac{π}{4} - x) ∣ + c$
$= log ∣ cosec (\frac{π}{4} + x) + cot (\frac{π}{4} + x) ∣ + c = log ∣ cot (\frac{x}{2} + \frac{π}{8}) ∣ + c = log ∣ tan (\frac{x}{2} - \frac{3 π}{8}) ∣ + c$

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