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

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Answer : I=-\frac{1}{2} \log \left|\cos e c\left(x+\frac{\pi}{3}\right)+c \operatorname{ot}\left(x+\frac{\pi}{3}\right)\right|

Hint: To solve this question we have to rewrite the formula of cosx and sinx by adding the terms

\sin \frac{\pi}{3} \operatorname{and} \cos \frac{\pi}{3}

Given : I=\int \frac{1}{\sin x+\sqrt{3} \cos x} d x

Solution :

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

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

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

Note : Final answer is not matching.

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