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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 1

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Answer: \log |x+\log(\sin x)|+c

Hint: You must know about the integration rule of sin, cot and logarithm function


\int \frac{1+\cot x}{x+\log\sin x}dx


Consider x+\log(\sin x)=t    and differentiate both sides,

We get (1+\cot x) dx=dt

By integrating w.r.t t

  \int \frac{dt}{t}=\log t +c                                                                                 \left [\int \frac{1}{x} dx=\log x+c \right ]                                 

By substituting the value of t

\int \frac{1+\cot x}{x+\log \sin x}dx

= \log |x+\log(\sin x)|+c

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