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Solving integrals/ x/(1-sqrt(x))

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

The solution

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  1             
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 |      x       
 |  --------- dx
 |        ___   
 |  1 - \/ x    
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0

$$\int\limits_{0}^{1} \frac{x}{1 - \sqrt{x}}\, dx$$

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

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                           
 |                                                         3/2
 |     x                      ___        /       ___\   2*x   
 | --------- dx = C - x - 2*\/ x  - 2*log\-1 + \/ x / - ------
 |       ___                                              3   
 | 1 - \/ x                                                   
 |                                                            
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo + 2*pi*I

$$\infty + 2 i \pi$$

oo + 2*pi*I

$$\infty + 2 i \pi$$

oo + 2*pi*i

Numerical answer [src]

85.9014007435013

85.9014007435013

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2