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Identical expressions
(three -ln*x)/x
(3 minus ln multiply by x) divide by x
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(3-lnx)/x
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(3-ln*x) divide by x
(3-ln*x)/xdx
Similar expressions
(3+ln*x)/x
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Integral of (3-ln*x)/x dx

The solution

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  E              
  /              
 |               
 |  3 - log(x)   
 |  ---------- dx
 |      x        
 |               
/                
 -1              
e

$$\int\limits_{e^{-1}}^{e} \frac{3 - \log{\left(x \right)}}{x}\, dx$$

Integral((3 - log(x))/x, (x, exp(-1), E))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 3 - \log{\left(x \right)}$ .
  
  Then let $du = - \frac{dx}{x}$ and substitute $- du$ :
  
  $\int \left(- u\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u\, du = - \int u\, du$
    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: $- \frac{u^{2}}{2}$
  Now substitute $u$ back in:
  
  $- \frac{\left(3 - \log{\left(x \right)}\right)^{2}}{2}$
Method #2
1. Rewrite the integrand:
  
  $\frac{3 - \log{\left(x \right)}}{x} = - \frac{\log{\left(x \right)} - 3}{x}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\log{\left(x \right)} - 3}{x}\right)\, dx = - \int \frac{\log{\left(x \right)} - 3}{x}\, dx$
  1. Let $u = \frac{1}{x}$ .
    
    Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
    
    $\int \left(- \frac{\log{\left(\frac{1}{u} \right)} - 3}{u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\log{\left(\frac{1}{u} \right)} - 3}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)} - 3}{u}\, du$
      
      Let $u = \log{\left(\frac{1}{u} \right)} - 3$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int \left(- u\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\left(\log{\left(\frac{1}{u} \right)} - 3\right)^{2}}{2}$
      
      So, the result is: $\frac{\left(\log{\left(\frac{1}{u} \right)} - 3\right)^{2}}{2}$
    Now substitute $u$ back in:
    
    $\frac{\left(\log{\left(x \right)} - 3\right)^{2}}{2}$
  So, the result is: $- \frac{\left(\log{\left(x \right)} - 3\right)^{2}}{2}$
Method #3
1. Rewrite the integrand:
  
  $\frac{3 - \log{\left(x \right)}}{x} = - \frac{\log{\left(x \right)}}{x} + \frac{3}{x}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\log{\left(x \right)}}{x}\right)\, dx = - \int \frac{\log{\left(x \right)}}{x}\, dx$
    1. Let $u = \frac{1}{x}$ .
      
      Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du$
      
      Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int \left(- u\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{2}$
      
      So, the result is: $\frac{\log{\left(\frac{1}{u} \right)}^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\log{\left(x \right)}^{2}}{2}$
    So, the result is: $- \frac{\log{\left(x \right)}^{2}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3}{x}\, dx = 3 \int \frac{1}{x}\, dx$
    1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
    So, the result is: $3 \log{\left(x \right)}$
  The result is: $- \frac{\log{\left(x \right)}^{2}}{2} + 3 \log{\left(x \right)}$
Now simplify:

$- \frac{\left(\log{\left(x \right)} - 3\right)^{2}}{2}$
Add the constant of integration:

$- \frac{\left(\log{\left(x \right)} - 3\right)^{2}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\left(\log{\left(x \right)} - 3\right)^{2}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                                 2
 | 3 - log(x)          (3 - log(x)) 
 | ---------- dx = C - -------------
 |     x                     2      
 |                                  
/

$$\int \frac{3 - \log{\left(x \right)}}{x}\, dx = C - \frac{\left(3 - \log{\left(x \right)}\right)^{2}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$6$$

Numerical answer [src]

6.0

6.0

Use the examples entering the upper and lower limits of integration.

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (3-ln*x)/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3