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Identical expressions
-(ln(x))/x- one /x-c
minus (ln(x)) divide by x minus 1 divide by x minus c
minus (ln(x)) divide by x minus one divide by x minus c
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-(ln(x))/x-1/x-cdx
Similar expressions
(ln(x))/x-1/x-c
-(ln(x))/x-1/x+c
-(ln(x))/x+1/x-c

Integral of -(ln(x))/x-1/x-c dx

The solution

You have entered [src]

  1                      
  /                      
 |                       
 |  /-log(x)    1    \   
 |  |-------- - - - c| dx
 |  \   x       x    /   
 |                       
/                        
0

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

Integral((-log(x))/x - 1/x - c, (x, 0, 1))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

oo - c

$$- c + \infty$$

oo - c

$$- c + \infty$$

oo - c

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

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Identical expressions

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Integral of -(ln(x))/x-1/x-c dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2