Mister Exam

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Identical expressions
ln^ five (x)/x
ln to the power of 5(x) divide by x
ln to the power of five (x) divide by x
ln5(x)/x
ln5x/x
ln⁵(x)/x
ln^5x/x
ln^5(x) divide by x
ln^5(x)/xdx

Solving integrals/ ln^5(x)/x

Integral of ln^5(x)/x dx

The solution

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  E           
  /           
 |            
 |     5      
 |  log (x)   
 |  ------- dx
 |     x      
 |            
/             
1

\int\limits_{1}^{e} \frac{\log{\left(x \right)}^{5}}{x}\, dx

Integral(log(x)^5/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 $du$ :
  
  $\int u^{5}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int u^{5}\, du = \frac{u^{6}}{6}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(x \right)}^{6}}{6}$
Method #2
1. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
  
  $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{5}}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\log{\left(\frac{1}{u} \right)}^{5}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{5}}{u}\, du$
    1. Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int \left(- u^{5}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{5}\, du = - \int u^{5}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{5}\, du = \frac{u^{6}}{6}$
      
      So, the result is: $- \frac{u^{6}}{6}$
      
      Now substitute $u$ back in:
      
      $- \frac{\log{\left(\frac{1}{u} \right)}^{6}}{6}$
    So, the result is: $\frac{\log{\left(\frac{1}{u} \right)}^{6}}{6}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(x \right)}^{6}}{6}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                        
 |                         
 |    5                6   
 | log (x)          log (x)
 | ------- dx = C + -------
 |    x                6   
 |                         
/

\int \frac{\log{\left(x \right)}^{5}}{x}\, dx = C + \frac{\log{\left(x \right)}^{6}}{6}

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/6

\frac{1}{6}

1/6

\frac{1}{6}

1/6

Numerical answer [src]

0.166666666666667

0.166666666666667

The graph

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

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How to use it?

Identical expressions

Integral of ln^5(x)/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2