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
Questions 203525
Subjects 9
Chapters 126
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