Mathematics

Evaluate the given integral.
$$\int { \log _{ 10 }{ x }  } dx$$


SOLUTION
$$I=\displaystyle\int{\log_{10}{x}\,dx}$$

Integrating by parts, we get

Let $$u=\log_{10}{x}\Rightarrow\,du=\log{10}\dfrac{1}{x}dx$$

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

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

$$I=x\log_{10}{x}-\dfrac{1}{\log{10}}\int{x\times\dfrac{1}{x}dx}$$

$$I=x\log_{10}{x}-\int{dx}$$

$$I=x\log_{10}{x}-x+c$$

$$\therefore\,I=\dfrac{1}{\log{10}}\left\{x\left(\log{x}-1\right)\right\}+c$$
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Subjective Medium Published on 17th 09, 2020
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