Mathematics

Evaluate :$$\displaystyle \int { \log } x{dx}$$


SOLUTION
Let $$I=\displaystyle\int \log x dx$$

Let $$y=\log x$$

$$\therefore x=e^y$$

$$\therefore$$ using integration by parts

$$I=\displaystyle\int y dx=xy-\displaystyle\int x dy$$

$$=x log x-\displaystyle\int e^ydy$$

$$=x log x -e^y$$

$$=x log x-e^{log x}+x$$                $$[\because e^{log^x_e}=x]$$

$$I=xlog x-x+c$$.
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Subjective Medium Published on 17th 09, 2020
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