First slide

Methods of integration

Question

The value of $\int \frac{1}{\sin (x - \frac{π}{3}) \cos x} d x$ , is

Moderate

A

$2 \log | \sin x + \sin (x - π / 3) | + C$
B

$2 \log |\sin x \sin (x - \frac{π}{3})| + C$
C

$2 \log |\sin x - \sin (x - \frac{π}{3})| + C$
D

none of these

Solution

We have,
$I = \int \frac{1}{\sin (x - \frac{π}{3}) \cos x} d x = \frac{1}{\cos \frac{π}{3}} \int \frac{\cos \{x - (x - \frac{π}{3})\}}{\sin (x - \frac{π}{3}) \cos x} d x$
$\begin{array}{l} \Rightarrow I = 2 \int \frac{\cos x \cos (x - \frac{π}{3}) + \sin x \sin (x - \frac{π}{3})}{\sin (x - \frac{π}{3}) \cos x} d x \\ \Rightarrow I = 2 \int \{\cot (x - \frac{π}{3}) + \tan x\} d x \\ \Rightarrow I = 2 \{\log |\sin (x - \frac{π}{3})| - \log | \cos x |\} + C \\ \Rightarrow I = 2 \log |\sin (x - \frac{π}{3}) \sec x| + C \end{array}$

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