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

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

Answers (1)

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

Solution:

\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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