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Integral of 1/((sqrtx-1)x^2) dx

The solution

You have entered [src]

 oo                  
  /                  
 |                   
 |        1          
 |  -------------- dx
 |  /  ___    \  2   
 |  \\/ x  - 1/*x    
 |                   
/                    
1

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{1}{x^{2} \left(\sqrt{x} - 1\right)} = - \frac{1}{- x^{\frac{5}{2}} + x^{2}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{- x^{\frac{5}{2}} + x^{2}}\right)\, dx = - \int \frac{1}{- x^{\frac{5}{2}} + x^{2}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\log{\left(x \right)} - 2 \log{\left(\sqrt{x} - 1 \right)} - \frac{1}{x} - \frac{2}{\sqrt{x}}$
  So, the result is: $- \log{\left(x \right)} + 2 \log{\left(\sqrt{x} - 1 \right)} + \frac{1}{x} + \frac{2}{\sqrt{x}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{1}{x^{2} \left(\sqrt{x} - 1\right)} = \frac{1}{x^{\frac{5}{2}} - x^{2}}$
2. Rewrite the integrand:
  
  $\frac{1}{x^{\frac{5}{2}} - x^{2}} = - \frac{1}{- x^{\frac{5}{2}} + x^{2}}$
3. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{- x^{\frac{5}{2}} + x^{2}}\right)\, dx = - \int \frac{1}{- x^{\frac{5}{2}} + x^{2}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\log{\left(x \right)} - 2 \log{\left(\sqrt{x} - 1 \right)} - \frac{1}{x} - \frac{2}{\sqrt{x}}$
  So, the result is: $- \log{\left(x \right)} + 2 \log{\left(\sqrt{x} - 1 \right)} + \frac{1}{x} + \frac{2}{\sqrt{x}}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

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

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Integral of 1/((sqrtx-1)x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2