Mathematics

# Write a value of $\displaystyle \int { \cfrac { \log { { x }^{ n } } }{ x } } dx$

##### SOLUTION
$I=\displaystyle\int{\dfrac{\log{{x}^{n}}}{x}dx}$

$=\displaystyle\int{\dfrac{n\log{x}}{x}dx}$

$=n\displaystyle\int{\dfrac{\log{x}}{x}dx}$

Let $t=\log{x}\Rightarrow\,dt=\dfrac{dx}{x}$

$=n\displaystyle\int{t\,dt}$

$=n\dfrac{{t}^{2}}{2}+c$     .....where $c$ is the constant of integration.

$=\dfrac{n}{2}{\left(\log{x}\right)}^{2}+c$  .....where $t=\log{x}$

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

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