Mathematics

Write a value of 
$$\int { \log _{ e }{ x }  } dx\quad $$


SOLUTION
$$I=\displaystyle\int{\log_{e}{x}\,dx}$$

 $$=\displaystyle\int{1.\log_{e}{x}\,dx}$$

Integrating by parts, we get

Let $$u=\log_{e}{x}\Rightarrow\,du=\dfrac{1}{x}dx$$

$$dv=dx\Rightarrow\,v=x$$

$$\int u.v dx=u \int vdx-\int \left [\int vdx. \dfrac{du}{dx}.dx \right ] $$......by parts formula.

$$\displaystyle\int{\log_{e}{x}\,dx}=x\log_{e}{x}-\displaystyle\int{x\times\dfrac{dx}{x}}$$

$$=x\log_{e}{x}-\displaystyle\int{dx}$$

$$=x\log_{e}{x}-x+c$$

$$=x\left(\log_{e}{x}-1\right)+c$$ where $$c$$ is the constant of integration.
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Subjective Medium Published on 17th 09, 2020
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