Mister Exam

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Identical expressions
one *(lnx/ two *lnx)*(one /x)
1 multiply by (lnx divide by 2 multiply by lnx) multiply by (1 divide by x)
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1(lnx/2lnx)(1/x)
1lnx/2lnx1/x
1*(lnx divide by 2*lnx)*(1 divide by x)
1*(lnx/2*lnx)*(1/x)dx

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

The solution

You have entered [src]

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

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

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

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 $\frac{du}{2}$ :
  
  $\int \frac{u^{2}}{2}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u^{2}\, du = \frac{\int u^{2}\, du}{2}$
    1. 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: $\frac{u^{3}}{6}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(x \right)}^{3}}{6}$
Method #2
1. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $- \frac{du}{2}$ :
  
  $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{2 u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du = - \frac{\int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du}{2}$
    1. Let $u = \frac{1}{u}$ .
      
      Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{\log{\left(u \right)}^{2}}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\log{\left(u \right)}^{2}}{u}\, du = - \int \frac{\log{\left(u \right)}^{2}}{u}\, du$
      
      Let $u = \log{\left(u \right)}$ .
      
      Then let $du = \frac{du}{u}$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\log{\left(u \right)}^{3}}{3}$
      
      So, the result is: $- \frac{\log{\left(u \right)}^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\log{\left(u \right)}^{3}}{3}$
    So, the result is: $- \frac{\log{\left(u \right)}^{3}}{6}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(x \right)}^{3}}{6}$
Add the constant of integration:

$\frac{\log{\left(x \right)}^{3}}{6}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(x \right)}^{3}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              
 |                               
 | log(x)                        
 | ------*log(x)             3   
 |   2                    log (x)
 | ------------- dx = C + -------
 |       x                   6   
 |                               
/

$$\int \frac{\frac{\log{\left(x \right)}}{2} \log{\left(x \right)}}{x}\, dx = C + \frac{\log{\left(x \right)}^{3}}{6}$$

Simplify

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

Numerical answer [src]

14284.1898578166

14284.1898578166

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

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How to use it?

Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Other calculators

How to use it?

Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

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