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

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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.


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