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explain solution RD Sharma class 12 chapter Indefinite Integrals exercise 18.9 question 28 maths

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Answer: (\log x)^{2}+c

Hint:Use substitution method to solve this integral.

Given:   \int \frac{\log x^{2}}{x} d x


        \text { Let } I=\int \frac{\log x^{2}}{x} d x

                    =\int \frac{2 \log x}{x} d x\left[\because \log x^{m}=m \log x\right]

        \operatorname{Put} \log x=t \Rightarrow \frac{1}{x} d x=d t, \text { then }

                I=\int 2 \cdot \frac{t}{x} \cdot x \; d t=2 \int t\; d t=2 \int \frac{t^{1+1}}{1+1}+c                    \left[\because \int x^{n} d x=\frac{x^{n+1}}{n+1}+c\right]

                   =2 \frac{t^{1+1}}{2}+c=t^{2}+c=(\log x)^{2}+c                                  [\because t=\log x]


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