Mister Exam

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

The solution

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

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

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

Detail solution

Let $u = \sqrt{x}$ .

Then let $du = \frac{dx}{2 \sqrt{x}}$ and substitute $2 du$ :

$\int \frac{2 u}{u - 1}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{u}{u - 1}\, du = 2 \int \frac{u}{u - 1}\, du$
  1. Rewrite the integrand:
    
    $\frac{u}{u - 1} = 1 + \frac{1}{u - 1}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    1. Let $u = u - 1$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      Now substitute $u$ back in:
      
      $\log{\left(u - 1 \right)}$
    The result is: $u + \log{\left(u - 1 \right)}$
  So, the result is: $2 u + 2 \log{\left(u - 1 \right)}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

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/(x^(1/2)-1) dx

Limits of integration:

The graph:

Piecewise:

The solution