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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

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$\int {{e^x}\left[ {{\mathop{\rm tanx}\nolimits} - log\left( {\cos x} \right)} \right]} dx =$
• A. ${e^x}\log \left( {co\sec x} \right) + c$
• B. ${e^x}\log \left( {\cos x} \right) + c$
• C. ${e^x}\log \left( {\sin x} \right) + c$
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1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \frac { 1 - x ^ { 2 } } { \left( 1 + x ^ { 2 } \right) \sqrt { 1 + x ^ { 4 } } } d x$ is equal to
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1 Verified Answer | Published on 17th 09, 2020

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