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Identical expressions
(sqrt(one -lnx))/x
( square root of (1 minus lnx)) divide by x
( square root of (one minus lnx)) divide by x
(√(1-lnx))/x
sqrt1-lnx/x
(sqrt(1-lnx)) divide by x
(sqrt(1-lnx))/xdx
Similar expressions
(sqrt(1+lnx))/x

Integral of (sqrt(1-lnx))/x dx

The solution

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

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 1 - \log{\left(x \right)}$ .
  
  Then let $du = - \frac{dx}{x}$ and substitute $- du$ :
  
  $\int \left(- \sqrt{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sqrt{u}\, du = - \int \sqrt{u}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
    So, the result is: $- \frac{2 u^{\frac{3}{2}}}{3}$
  Now substitute $u$ back in:
  
  $- \frac{2 \left(1 - \log{\left(x \right)}\right)^{\frac{3}{2}}}{3}$
Method #2
1. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
  
  $\int \left(- \frac{\sqrt{1 - \log{\left(\frac{1}{u} \right)}}}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sqrt{1 - \log{\left(\frac{1}{u} \right)}}}{u}\, du = - \int \frac{\sqrt{1 - \log{\left(\frac{1}{u} \right)}}}{u}\, du$
    1. Let $u = 1 - \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = \frac{du}{u}$ and substitute $du$ :
      
      $\int \sqrt{u}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{2 \left(1 - \log{\left(\frac{1}{u} \right)}\right)^{\frac{3}{2}}}{3}$
    So, the result is: $- \frac{2 \left(1 - \log{\left(\frac{1}{u} \right)}\right)^{\frac{3}{2}}}{3}$
  Now substitute $u$ back in:
  
  $- \frac{2 \left(1 - \log{\left(x \right)}\right)^{\frac{3}{2}}}{3}$
Add the constant of integration:

$- \frac{2 \left(1 - \log{\left(x \right)}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{2 \left(1 - \log{\left(x \right)}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                                          
 |   ____________                        3/2
 | \/ 1 - log(x)           2*(1 - log(x))   
 | -------------- dx = C - -----------------
 |       x                         3        
 |                                          
/

$$\int \frac{\sqrt{1 - \log{\left(x \right)}}}{x}\, dx = C - \frac{2 \left(1 - \log{\left(x \right)}\right)^{\frac{3}{2}}}{3}$$

Simplify

The answer [src]

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$$\infty$$

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$$\infty$$

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Numerical answer [src]

201.185008592488

201.185008592488

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

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Identical expressions

Similar expressions

Integral of (sqrt(1-lnx))/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2