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Formula
Calculate the integral
Answer
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$\int{ \dfrac{ \ln{\left( x \right)} }{ x } }d{ x }$
$\dfrac { 1 } { 2 } \ln { \left( x \right) } ^ { 2 } + C$
Calculate the indefinite integral.
$\displaystyle\int { \color{#FF6800}{ \dfrac { \ln { \left( x \right) } } { x } } } d { \color{#FF6800}{ x } }$
$ $ Calculate the integral of the logarithmic function using the formula of $ \int x^{-1} \ln(x)^{n} dx = \left[\int{u^{n}}d{u}\right]_{u=\ln{\left(x\right)}}$
$\left [ \displaystyle\int { \color{#FF6800}{ u } } d { \color{#FF6800}{ u } } \right ] _ { \color{#FF6800}{ u } = \ln { \left( \color{#FF6800}{ x } \right) } }$
$\left [ \displaystyle\int { \color{#FF6800}{ u } } d { \color{#FF6800}{ u } } \right ] _ { u = \ln { \left( x \right) } }$
$ $ Calculate the integral using the formula of $ \int{x^{n}}dx = \frac{x^{n+1}}{n+1}$
$\left [ \color{#FF6800}{ \frac { 1 } { 1 + 1 } } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \right ] _ { u = \ln { \left( x \right) } }$
$\left [ \frac { 1 } { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } u ^ { 1 + 1 } \right ] _ { u = \ln { \left( x \right) } }$
$ $ Add $ 1 $ and $ 1$
$\left [ \frac { 1 } { \color{#FF6800}{ 2 } } u ^ { 1 + 1 } \right ] _ { u = \ln { \left( x \right) } }$
$\left [ \frac { 1 } { 2 } u ^ { \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } \right ] _ { u = \ln { \left( x \right) } }$
$ $ Add $ 1 $ and $ 1$
$\left [ \frac { 1 } { 2 } u ^ { \color{#FF6800}{ 2 } } \right ] _ { u = \ln { \left( x \right) } }$
$\left [ \color{#FF6800}{ \frac { 1 } { 2 } } \color{#FF6800}{ u } ^ { \color{#FF6800}{ 2 } } \right ] _ { \color{#FF6800}{ u } = \ln { \left( \color{#FF6800}{ x } \right) } }$
$ $ Return the substituted value $ $
$\color{#FF6800}{ \dfrac { 1 } { 2 } } \ln { \left( \color{#FF6800}{ x } \right) } ^ { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ \dfrac { 1 } { 2 } } \ln { \left( \color{#FF6800}{ x } \right) } ^ { \color{#FF6800}{ 2 } }$
$ $ Add the integral constant $ C $ . $ $
$\color{#FF6800}{ \dfrac { 1 } { 2 } } \ln { \left( \color{#FF6800}{ x } \right) } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ C }$
Solution search results
search-thumbnail-If the sum of two consecutive 
numbers is $45$ and one number is $X$ 
.This statement in the form of 
equation $1s:$ 
$\left(1$ Point) $\right)$ 
$○5x+1$ $1eft\left(x+1$ $r1gnt\right)=45s$ 
$○sx+1ef\left(x+2$ $r1gnt\right)=145s$ 
$sx+1x=45s$
7th-9th grade
Algebra
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