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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 24

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Answer:\log|\log(\sin x)|+c

Hints: You must know about the integral rule of logarithm and trigonometric functions

Given: \int \frac{\cos x}{\sin x\log \sin x}dx

Solution:\log \sin x =t     and differentiate both sides, \left [ \frac{d}{dx} \log \sin x=\frac{1}{\sin x}.\frac{d}{dx}\sin x\right ]

\begin{aligned} &\frac{1}{\sin x} \cdot \cos x=d t \\ &\int \frac{\cos x d x}{\sin x \cdot \log \sin x}=\int \frac{d t}{t} \\ &\log |t|+c \\ &\log |\log (\sin x)|+c \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left[\int \frac{1}{x} d x=\log x+c\right] \end{aligned}   

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