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five \x(lnx+ two)
5\x(lnx plus 2)
five \x(lnx plus two)
5\xlnx+2
5\x(lnx+2)dx
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5\x(lnx-2)

Integral of 5\x(lnx+2) dx

The solution

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  1                  
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 |  5                
 |  -*(log(x) + 2) dx
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0

$$\int\limits_{0}^{1} \frac{5}{x} \left(\log{\left(x \right)} + 2\right)\, dx$$

Integral((5/x)*(log(x) + 2), (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 $- du$ :
  
  $\int \left(- \frac{5 \log{\left(\frac{1}{u} \right)} + 10}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{5 \log{\left(\frac{1}{u} \right)} + 10}{u}\, du = - \int \frac{5 \log{\left(\frac{1}{u} \right)} + 10}{u}\, du$
    1. Let $u = \frac{1}{u}$ .
      
      Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{5 \log{\left(u \right)} + 10}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{5 \log{\left(u \right)} + 10}{u}\, du = - \int \frac{5 \log{\left(u \right)} + 10}{u}\, du$
      
      Let $u = \log{\left(u \right)}$ .
      
      Then let $du = \frac{du}{u}$ and substitute $du$ :
      
      $\int \left(5 u + 10\right)\, du$
      
      Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 5 u\, du = 5 \int u\, du$
      
      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: $\frac{5 u^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 10\, du = 10 u$
      
      The result is: $\frac{5 u^{2}}{2} + 10 u$
      
      Now substitute $u$ back in:
      
      $\frac{5 \log{\left(u \right)}^{2}}{2} + 10 \log{\left(u \right)}$
      
      So, the result is: $- \frac{5 \log{\left(u \right)}^{2}}{2} - 10 \log{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- \frac{5 \log{\left(u \right)}^{2}}{2} + 10 \log{\left(u \right)}$
    So, the result is: $\frac{5 \log{\left(u \right)}^{2}}{2} - 10 \log{\left(u \right)}$
  Now substitute $u$ back in:
  
  $\frac{5 \log{\left(x \right)}^{2}}{2} + 10 \log{\left(x \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{5}{x} \left(\log{\left(x \right)} + 2\right) = \frac{5 \log{\left(x \right)} + 10}{x}$
2. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
  
  $\int \left(- \frac{5 \log{\left(\frac{1}{u} \right)} + 10}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{5 \log{\left(\frac{1}{u} \right)} + 10}{u}\, du = - \int \frac{5 \log{\left(\frac{1}{u} \right)} + 10}{u}\, du$
    1. Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $du$ :
      
      $\int \left(- 5 u - 10\right)\, du$
      
      Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 5 u\right)\, du = - 5 \int u\, du$
      
      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: $- \frac{5 u^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \left(-10\right)\, du = - 10 u$
      
      The result is: $- \frac{5 u^{2}}{2} - 10 u$
      
      Now substitute $u$ back in:
      
      $- \frac{5 \log{\left(\frac{1}{u} \right)}^{2}}{2} - 10 \log{\left(\frac{1}{u} \right)}$
    So, the result is: $\frac{5 \log{\left(\frac{1}{u} \right)}^{2}}{2} + 10 \log{\left(\frac{1}{u} \right)}$
  Now substitute $u$ back in:
  
  $\frac{5 \log{\left(x \right)}^{2}}{2} + 10 \log{\left(x \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{5}{x} \left(\log{\left(x \right)} + 2\right) = \frac{5 \log{\left(x \right)}}{x} + \frac{10}{x}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{5 \log{\left(x \right)}}{x}\, dx = 5 \int \frac{\log{\left(x \right)}}{x}\, dx$
    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: $\frac{5 \log{\left(x \right)}^{2}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{10}{x}\, dx = 10 \int \frac{1}{x}\, dx$
    1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
    So, the result is: $10 \log{\left(x \right)}$
  The result is: $\frac{5 \log{\left(x \right)}^{2}}{2} + 10 \log{\left(x \right)}$
Now simplify:

$\frac{5 \left(\log{\left(x \right)} + 4\right) \log{\left(x \right)}}{2}$
Add the constant of integration:

$\frac{5 \left(\log{\left(x \right)} + 4\right) \log{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{5 \left(\log{\left(x \right)} + 4\right) \log{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                                          2   
 | 5                                   5*log (x)
 | -*(log(x) + 2) dx = C + 10*log(x) + ---------
 | x                                       2    
 |                                              
/

$$\int \frac{5}{x} \left(\log{\left(x \right)} + 2\right)\, dx = C + \frac{5 \log{\left(x \right)}^{2}}{2} + 10 \log{\left(x \right)}$$

Simplify

The answer [src]

-oo

$$-\infty$$

-oo

$$-\infty$$

-oo

Numerical answer [src]

-4418.91485573671

-4418.91485573671

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

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Identical expressions

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Integral of 5\x(lnx+2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3