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

Solving integrals/ x/x√1-ln^2x

Integral of x/x√1-ln^2x dx

The solution

You have entered [src]

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

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

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

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:
  
  $\int \frac{\sqrt{1} x}{x}\, dx = \int \frac{x}{x}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $x$
  So, the result is: $x$
The result is: $- x \log{\left(x \right)}^{2} + 2 x \log{\left(x \right)} - x$
Now simplify:

$x \left(- \log{\left(x \right)}^{2} + 2 \log{\left(x \right)} - 1\right)$
Add the constant of integration:

$x \left(- \log{\left(x \right)}^{2} + 2 \log{\left(x \right)} - 1\right)+ \mathrm{constant}$

The answer is:

x \left(- \log{\left(x \right)}^{2} + 2 \log{\left(x \right)} - 1\right)+ \mathrm{constant}

The answer (Indefinite) [src]

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

x-x\,\left(\left(\log x\right)^2-2\,\log x+2\right)

Simplify

The graph

Plot the graph f(x)

The answer [src]

      ___     ___    2/  ___\       ___    /  ___\
1 - \/ 2  - \/ 2 *log \\/ 2 / + 2*\/ 2 *log\\/ 2 /

-{{\left(\log 2\right)^2-4\,\log 2-2^{{{3}\over{2}}}+4}\over{2^{{{3 }\over{2}}}}}

      ___     ___    2/  ___\       ___    /  ___\
1 - \/ 2  - \/ 2 *log \\/ 2 / + 2*\/ 2 *log\\/ 2 /

- \sqrt{2} - \sqrt{2} \log{\left(\sqrt{2} \right)}^{2} + 2 \sqrt{2} \log{\left(\sqrt{2} \right)} + 1

Упростить

Numerical answer [src]

0.396178789003915

0.396178789003915

The graph

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

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Integral of x/x√1-ln^2x dx

Limits of integration:

The graph:

Piecewise:

The solution