Question

Evaluate: $\int \frac{dx}{\cos x-\sin x}$

• A. $\frac{1}{\sqrt{2}}\log \left|\begin{array}{c}\tan (\frac{x}{2}+\frac{3\pi }{8})\end{array}\right|+c$
• B. $\frac{1}{\sqrt{2}}\log \left|\begin{array}{c}\cot \left(\frac{x}{2}\right)\end{array}\right|+c$
• C. $\frac{1}{\sqrt{2}}\log \left|\begin{array}{c}\tan (\frac{x}{2}-\frac{3\pi }{8})\end{array}\right|+c$
• D. $\frac{1}{\sqrt{2}}\log \left|\begin{array}{c}(\frac{x}{2}+\frac{3\pi }{8})\end{array}\right|+c$

Answer 1

The correct option is A $\frac{1}{\sqrt{2}}\log \left|\begin{array}{c}\tan (\frac{x}{2}+\frac{3\pi }{8})\end{array}\right|+c$

We are given,

$I=\int \frac{dx}{\cos x-\sin x}$

$=\frac{1}{\sqrt{2}}\int \frac{dx}{\frac{1}{\sqrt{2}}\cos x-\frac{1}{\sqrt{2}}\sin x}$

(multiply & divide I with $1/\sqrt{2}$)

$I=\frac{1}{\sqrt{2}}\int \frac{dx}{\cos x\cos \left(\frac{\pi }{4}\right)-\sin x\sin \left(\pi /4\right)}$

$I=\frac{1}{\sqrt{2}}\int \frac{dx}{\cos (x+\frac{\pi }{4})}$

$=\frac{1}{\sqrt{2}}\int \sec (x+\frac{\pi }{4})dx$

(we know that $\cos (A+B)=\cos A\cos B-\sin A\sin B)$

$I=\frac{1}{\sqrt{2}}log\left|\tan (\frac{x}{2}+\frac{\pi }{8}+\frac{\pi }{4})\right|+C$

$I=\frac{1}{\sqrt{2}}log\left|\tan (\frac{x}{2}+\frac{3\pi }{8})\right|+C$.