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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 9

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Answer: \frac{n}{2}\left ( \log x \right )^{2}+c

Hint: You must know about the integration rule of logarithm function

Given: \int \frac{\log x^{n}}{x}dx


\int \frac{\log x^{n}}{x}dx

= \int \frac{n\log x}{x}dx                                                                                         [\because \log x^{a}=a\log x]

Put \log x=t   and differentiate both sides,  \frac{d}{dx}\log x= \frac{1}{x}

\begin{aligned} &\frac{1}{x} d x=d t \\ &I=\int n(t) d t \\ &=n\left(\frac{t^{2}}{2}+c\right) \quad\quad \quad \quad \quad \quad \quad \quad \quad \quad \left [ \int x^{n}dx=\frac{x^{n+1}}{n+1} \right ] \\ &=\frac{n(\log x)^{2}}{2}+n c \end{aligned}

= \frac{n}{2}\left ( \log x \right )^{2}+c                                                                                    

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