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dx/x√ one -ln^2x
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Similar expressions
dx/x√1+ln^2x

Integral of dx/x√1-ln^2x dx

The solution

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  E                     
  /                     
 |                      
 |  /  ___          \   
 |  |\/ 1       2   |   
 |  |----- - log (x)| dx
 |  \  x            /   
 |                      
/                       
1

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

Integral(sqrt(1)/x - log(x)^2, (x, 1, E))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \log{\left(x \right)}^{2}\right)\, dx = - \int \log{\left(x \right)}^{2}\, dx$
  1. Let $u = \log{\left(x \right)}$ .
    
    Then let $du = \frac{dx}{x}$ and substitute $du$ :
    
    $\int u^{2} e^{u}\, du$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = u^{2}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 2 u$ .
      
      To find $v{\left(u \right)}$ :
      1. The integral of the exponential function is itself.
        
        $\int e^{u}\, du = e^{u}$
      Now evaluate the sub-integral.
    2. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = 2 u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 2$ .
      
      To find $v{\left(u \right)}$ :
      1. The integral of the exponential function is itself.
        
        $\int e^{u}\, du = e^{u}$
      Now evaluate the sub-integral.
    3. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 e^{u}\, du = 2 \int e^{u}\, du$
      1. The integral of the exponential function is itself.
        
        $\int e^{u}\, du = e^{u}$
      So, the result is: $2 e^{u}$
    Now substitute $u$ back in:
    
    $x \log{\left(x \right)}^{2} - 2 x \log{\left(x \right)} + 2 x$
  So, the result is: $- x \log{\left(x \right)}^{2} + 2 x \log{\left(x \right)} - 2 x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\sqrt{1}}{x}\, dx = \int \frac{1}{x}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\log{\left(x \right)}$
  So, the result is: $\log{\left(x \right)}$
The result is: $- x \log{\left(x \right)}^{2} + 2 x \log{\left(x \right)} - 2 x + \log{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

3 - E

3 - e

3 - E

3 - e

3 - E

Numerical answer [src]

0.281718171540955

0.281718171540955

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

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Integral of dx/x√1-ln^2x dx

Limits of integration:

The graph:

Piecewise:

The solution