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one /x*(two - three *lnx*lnx)
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1/x*(2-3*lnx*lnx)dx
Similar expressions
1/x*(2+3*lnx*lnx)

Integral of 1/x(2-3lnx*lnx) dx

The solution

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  1                       
  /                       
 |                        
 |  2 - 3*log(x)*log(x)   
 |  ------------------- dx
 |           x            
 |                        
/                         
0

$$\int\limits_{0}^{1} \frac{- \log{\left(x \right)} 3 \log{\left(x \right)} + 2}{x}\, dx$$

Integral((2 - 3*log(x)*log(x))/x, (x, 0, 1))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

-oo

$$-\infty$$

-oo

$$-\infty$$

-oo

Numerical answer [src]

-85616.9582546317

-85616.9582546317

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of 1/x(2-3lnx*lnx) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of 1/x*(2-3*lnx*lnx) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3

Integral of 1/x(2-3lnx*lnx) dx