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Identical expressions
two (lnx+ one)/x
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two (lnx plus one) divide by x
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2(lnx+1) divide by x
2(lnx+1)/xdx
Similar expressions
2ln(x+1)/x
2(lnx-1)/x

Integral of 2(lnx+1)/x dx

The solution

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  2                  
 e                   
  /                  
 |                   
 |  2*(log(x) + 1)   
 |  -------------- dx
 |        x          
 |                   
/                    
E

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

Integral((2*(log(x) + 1))/x, (x, E, exp(2)))

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 $du$ :
  
  $\int \left(2 u + 2\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 u\, du = 2 \int u\, du$
      
      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: $u^{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 2\, du = 2 u$
    The result is: $u^{2} + 2 u$
  Now substitute $u$ back in:
  
  $\log{\left(x \right)}^{2} + 2 \log{\left(x \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{2 \left(\log{\left(x \right)} + 1\right)}{x} = \frac{2 \log{\left(x \right)}}{x} + \frac{2}{x}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2 \log{\left(x \right)}}{x}\, dx = 2 \int \frac{\log{\left(x \right)}}{x}\, dx$
    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: $\log{\left(x \right)}^{2}$
  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: $\log{\left(x \right)}^{2} + 2 \log{\left(x \right)}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                          
 |                                           
 | 2*(log(x) + 1)             2              
 | -------------- dx = C + log (x) + 2*log(x)
 |       x                                   
 |                                           
/

$$\int \frac{2 \left(\log{\left(x \right)} + 1\right)}{x}\, dx = C + \log{\left(x \right)}^{2} + 2 \log{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$5$$

Numerical answer [src]

5.0

5.0

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

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How to use it?

Identical expressions

Similar expressions

Integral of 2(lnx+1)/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2