Mathematics

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

##### SOLUTION
$\displaystyle\int{\log{x}dx}$

let $u=\log{x}$ $v=1$

Integrating by parts,
$=x\log{x}-\displaystyle\int{x\times\dfrac{1}{x}dx}$

$=x\log{x}-\displaystyle\int 1{dx}$

$=x\log{x}-x=x\left(\log{x}-1\right)+c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Medium
Evaluate: $\displaystyle \int _{ 0 }^{ 1 }{ { x\left( 1-x \right) }^{ n } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
If $\displaystyle f\left( x \right)=\int { \frac { { x }^{ 2 }dx }{ \left( 1+{ x }^{ 2 } \right) \left( 1+\sqrt { 1+{ x }^{ 2 } } \right) } }$ and $f\left( 0 \right) =0$, then the value of $f(1)$ is
• A. $\log { \left( 1+\sqrt { 2 } \right) }$
• B. $\displaystyle \log { \left( 1+\sqrt { 2 } \right) } +\frac { \pi }{ 2 }$
• C. None of these
• D. $\displaystyle \log { \left( 1+\sqrt { 2 } \right) } -\frac { \pi }{ 4 }$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Evaluate:
$\displaystyle\int _{ 0 }^{ \pi /2 }{ \log { \left( \sin { x } \right) dx } }$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate $\displaystyle \int_{1}^{3}(2x^2+5x)dx$

$\underset {n\rightarrow \infty}{lim}\dfrac{1^2+2^2+3^2+.....+n^2}{n^3}=.................$