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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 27

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Answer: \log|3+x\log x|+c

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

Given: \int \frac{1+\log x}{3+x\log x}dx


\int \frac{1+\log x}{3+x\log x}dx

Lett=3+x\log x and differentiate both sides

dt=0+x\frac{d}{dx}\left ( \log x \right )+\log x\frac{d}{dx}\left ( x \right )

dt=0+x.\frac{1}{x}+\left ( \log x \right )

Multiplication rule of differentiation \left [ \frac{d}{dx}uv=u\frac{d\left ( v \right )}{dx}+v\frac{d}{dx}(u) \right ]

dt=\left ( 1+\log x \right )dx 

I=\int \frac{1+\log x}{3+x\log x}dx

I=\int \frac{dt}{t}

=\log \left | t \right |+c

\therefore I=\log\left | 3+x\log x \right |+c                          \left [ \int \frac{1}{x}dx=\log x+c \right ]

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