Integral of loge(x) dx

The solution

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

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

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

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\log{\left(x \right)}}{\log{\left(e^{1} \right)}}\, dx = \frac{\int \log{\left(x \right)}\, dx}{\log{\left(e^{1} \right)}}$
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \log{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
  Now evaluate the sub-integral.
2. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
So, the result is: $\frac{x \log{\left(x \right)} - x}{\log{\left(e^{1} \right)}}$
Now simplify:

$x \left(\log{\left(x \right)} - 1\right)$
Add the constant of integration:

$x \left(\log{\left(x \right)} - 1\right)+ \mathrm{constant}$

The answer is:

$x \left(\log{\left(x \right)} - 1\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              
 |                               
 |  log(x)          -x + x*log(x)
 | ------- dx = C + -------------
 |    / 1\                / 1\   
 | log\e /             log\e /   
 |                               
/

$$\int \frac{\log{\left(x \right)}}{\log{\left(e^{1} \right)}}\, dx = C + \frac{x \log{\left(x \right)} - x}{\log{\left(e^{1} \right)}}$$

Simplify

The graph

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The answer [src]

$$1$$

Numerical answer [src]

1.0

1.0

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

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Integral of loge(x) dx

Limits of integration:

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The solution