Integral of dx/xln4x dx

The solution

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  1            
  /            
 |             
 |  log(4*x)   
 |  -------- dx
 |     x       
 |             
/              
0

$$\int\limits_{0}^{1} \frac{\log{\left(4 x \right)}}{x}\, dx$$

Integral(log(4*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 $- du$ :
  
  $\int \left(- \frac{\log{\left(\frac{1}{u} \right)} + 2 \log{\left(2 \right)}}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\log{\left(\frac{1}{u} \right)} + 2 \log{\left(2 \right)}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)} + 2 \log{\left(2 \right)}}{u}\, du$
    1. Let $u = \frac{1}{u}$ .
      
      Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{\log{\left(u \right)} + 2 \log{\left(2 \right)}}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\log{\left(u \right)} + 2 \log{\left(2 \right)}}{u}\, du = - \int \frac{\log{\left(u \right)} + 2 \log{\left(2 \right)}}{u}\, du$
      
      Let $u = \log{\left(u \right)} + 2 \log{\left(2 \right)}$ .
      
      Then let $du = \frac{du}{u}$ and substitute $du$ :
      
      $\int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\left(\log{\left(u \right)} + 2 \log{\left(2 \right)}\right)^{2}}{2}$
      
      So, the result is: $- \frac{\left(\log{\left(u \right)} + 2 \log{\left(2 \right)}\right)^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\left(- \log{\left(u \right)} + 2 \log{\left(2 \right)}\right)^{2}}{2}$
    So, the result is: $\frac{\left(- \log{\left(u \right)} + 2 \log{\left(2 \right)}\right)^{2}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\left(\log{\left(x \right)} + 2 \log{\left(2 \right)}\right)^{2}}{2}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\log{\left(4 x \right)}}{x} = \frac{\log{\left(x \right)} + 2 \log{\left(2 \right)}}{x}$
2. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
  
  $\int \left(- \frac{\log{\left(\frac{1}{u} \right)} + 2 \log{\left(2 \right)}}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\log{\left(\frac{1}{u} \right)} + 2 \log{\left(2 \right)}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)} + 2 \log{\left(2 \right)}}{u}\, du$
    1. Let $u = \log{\left(\frac{1}{u} \right)} + 2 \log{\left(2 \right)}$ .
      
      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:
      
      $- \frac{\left(\log{\left(\frac{1}{u} \right)} + 2 \log{\left(2 \right)}\right)^{2}}{2}$
    So, the result is: $\frac{\left(\log{\left(\frac{1}{u} \right)} + 2 \log{\left(2 \right)}\right)^{2}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\left(\log{\left(x \right)} + 2 \log{\left(2 \right)}\right)^{2}}{2}$
Method #3
1. Rewrite the integrand:
  
  $\frac{\log{\left(4 x \right)}}{x} = \frac{\log{\left(x \right)}}{x} + \frac{2 \log{\left(2 \right)}}{x}$
2. Integrate term-by-term:
  1. Let $u = \frac{1}{x}$ .
    
    Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
    
    $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}}{u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du$
      
      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:
    
    $\frac{\log{\left(x \right)}^{2}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2 \log{\left(2 \right)}}{x}\, dx = 2 \log{\left(2 \right)} \int \frac{1}{x}\, dx$
    1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
    So, the result is: $2 \log{\left(2 \right)} \log{\left(x \right)}$
  The result is: $\frac{\log{\left(x \right)}^{2}}{2} + 2 \log{\left(2 \right)} \log{\left(x \right)}$
Now simplify:

$\frac{\log{\left(4 x \right)}^{2}}{2}$
Add the constant of integration:

$\frac{\log{\left(4 x \right)}^{2}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(4 x \right)}^{2}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                      
 |                                      2
 | log(4*x)          (2*log(2) + log(x)) 
 | -------- dx = C + --------------------
 |    x                       2          
 |                                       
/

$$\int \frac{\log{\left(4 x \right)}}{x}\, dx = C + \frac{\left(\log{\left(x \right)} + 2 \log{\left(2 \right)}\right)^{2}}{2}$$

Simplify

The answer [src]

-oo

$$-\infty$$

-oo

$$-\infty$$

-oo

Numerical answer [src]

-910.841526560512

-910.841526560512

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

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Identical expressions

Integral of dx/xln4x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3