Integral of 4lnx/x dx

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  E            
  /            
 |             
 |  4*log(x)   
 |  -------- dx
 |     x       
 |             
/              
1

$$\int\limits_{1}^{e} \frac{4 \log{\left(x \right)}}{x}\, dx$$

Integral((4*log(x))/x, (x, 1, E))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \log{\left(x \right)}$ .
  
  Then let $du = \frac{dx}{x}$ and substitute $4 du$ :
  
  $\int 4 u\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u\, du = 4 \int u\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    So, the result is: $2 u^{2}$
  Now substitute $u$ back in:
  
  $2 \log{\left(x \right)}^{2}$
Method #2
1. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $- 4 du$ :
  
  $\int \left(- \frac{4 \log{\left(\frac{1}{u} \right)}}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du = - 4 \int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du$
    1. Let $u = \frac{1}{u}$ .
      
      Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{\log{\left(u \right)}}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\log{\left(u \right)}}{u}\, du = - \int \frac{\log{\left(u \right)}}{u}\, du$
      
      Let $u = \log{\left(u \right)}$ .
      
      Then let $du = \frac{du}{u}$ and substitute $du$ :
      
      $\int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\log{\left(u \right)}^{2}}{2}$
      
      So, the result is: $- \frac{\log{\left(u \right)}^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\log{\left(u \right)}^{2}}{2}$
    So, the result is: $2 \log{\left(u \right)}^{2}$
  Now substitute $u$ back in:
  
  $2 \log{\left(x \right)}^{2}$
Add the constant of integration:

$2 \log{\left(x \right)}^{2}+ \mathrm{constant}$

The answer is:

$2 \log{\left(x \right)}^{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                           
 |                            
 | 4*log(x)               2   
 | -------- dx = C + 2*log (x)
 |    x                       
 |                            
/

$$\int \frac{4 \log{\left(x \right)}}{x}\, dx = C + 2 \log{\left(x \right)}^{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$2$$

=

$$2$$

Numerical answer [src]

2.0

2.0

Other calculators

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Identical expressions

Similar expressions

Integral of 4lnx/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2