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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 20

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Hint: You must know about the integral rule of logarithm functions.

Given: \int \log_{e}xdx


\int \log_{e}x.1dx

Using by parts

         \begin{aligned} &=\log _{e} x \int 1 d x-\int\left[\frac{d}{d x} \log _{\varepsilon} x \int 1 d x\right] d x \\ &=x \log _{e} x-\int \frac{1}{x} x d x \\ &=x \log _{\varepsilon} x-\int 1 d x \quad\quad\quad\quad\quad\quad\left [ \int x^{n} dx=\frac{x^{n+1}}{n+1}\right ]\\ &=x \log _{e} x-x+c \\ &=x\left(\log _{e} x-1\right)+c \end{aligned}                                      


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