∫dxsin(x−π3)cosx=

∫dxsin(x−π3)cosx=

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$\int \frac{d x}{\sin (x - \frac{π}{3}) \cos x} =$

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2 [\log_{e} | \sin (x - π) . \sec x |] + c

2 [\log_{e} | \sin (x - \frac{π}{2}) . \sec x |] + c

2 [\log_{e} | \cos (x - \frac{π}{3}) . \sec x |] + c

2 [\log_{e} | \sin (x - \frac{π}{3}) . \sec x |] + c

answer is D.

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Detailed Solution

\int \frac{d x}{\sin (x - \frac{π}{3}) \cos x}

$= 2 \int \frac{\frac{1}{2}}{\sin (x - \frac{π}{3}) \cos x} d x$

$= 2 \int \frac{\cos [x - (x - \frac{π}{3})]}{\sin (x - \frac{π}{3}) \cos x} d x$

$= 2 \int \frac{\cos x \cos (x - \frac{π}{3}) + \sin x \sin (x - \frac{π}{3})}{\sin (x - \frac{π}{3}) \cos x} . d x$

$= 2 \int [\cot (x - \frac{π}{3}) + \tan x] d x$

$= 2 [\log_{e} | \sin (x - \frac{π}{3}) | + \log_{e} | \sec x |] + c$

$= 2 [\log_{e} | \sin (x - \frac{π}{3}) . \sec x |] + c$ .

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$\int \frac{d x}{\sin (x - \frac{π}{3}) \cos x} =$