#### Need Solution For  RD Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.23 Question 14 Maths Textbook Solution.

Answer : $=-\frac{1}{2} \log \left|\operatorname{cosec}\left(x-\frac{\pi}{3}\right)+\cot \left(x-\frac{\pi}{3}\right)\right|+c$

Hint : To solve this question we have to use formula of $\sin \left(x-\frac{\pi}{3}\right)$

Given : $\int \frac{1}{\sin x-\sqrt{3} \cos x} d x$

Solution : $\int \frac{1}{\sin x-\sqrt{3} \cos x} dx$

\begin{aligned} &=\int \frac{1}{2\left(\frac{1}{2} \sin x-\frac{\sqrt{3}}{2} \cos x\right)} d x \\ &=\frac{1}{2} \int \frac{1}{\sin x \cos \frac{\pi}{3}-\cos x \sin \frac{\pi}{3}} d x \end{aligned}

\begin{aligned} &=\frac{1}{2} \int \frac{1}{\sin \left(x-\frac{\pi}{3}\right)} d x \\ &=\frac{1}{2} \int \operatorname{cosec}\left(x-\frac{\pi}{3}\right) d x \\ &t=\left(x-\frac{\pi}{3}\right) \end{aligned}

\begin{aligned} &\frac{d t}{d x}=1 \\ &d t=d x \\ &=-\frac{1}{2} \log |\cos e c t+\cos t|+c \\ &=-\frac{1}{2} \log \left|\cos e c\left(x-\frac{\pi}{3}\right)+\cot \left(x-\frac{\pi}{3}\right)\right|+c \end{aligned}

Note: Final answer is not matching with the book.