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(x-2)/((x-1)(1/2))
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(x-2)/((x-1)^(1/2))dx
Similar expressions
(x+2)/((x-1)^(1/2))
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Solving integrals/ (x-2)/((x-1)^(1/2))

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

The solution

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  2             
  /             
 |              
 |    x - 2     
 |  --------- dx
 |    _______   
 |  \/ x - 1    
 |              
/               
1

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt{x - 1}$ .
  
  Then let $du = \frac{dx}{2 \sqrt{x - 1}}$ and substitute $du$ :
  
  $\int \left(2 u^{2} - 2\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 u^{2}\, du = 2 \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $\frac{2 u^{3}}{3}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int \left(-2\right)\, du = - 2 u$
    The result is: $\frac{2 u^{3}}{3} - 2 u$
  Now substitute $u$ back in:
  
  $\frac{2 \left(x - 1\right)^{\frac{3}{2}}}{3} - 2 \sqrt{x - 1}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x - 2}{\sqrt{x - 1}} = \frac{x}{\sqrt{x - 1}} - \frac{2}{\sqrt{x - 1}}$
2. Integrate term-by-term:
  1. Let $u = \frac{1}{\sqrt{x - 1}}$ .
    
    Then let $du = - \frac{dx}{2 \left(x - 1\right)^{\frac{3}{2}}}$ and substitute $du$ :
    
    $\int \left(- 2 \left(1 + \frac{1}{u^{2}}\right)^{2} + 2 + \frac{2}{u^{2}}\right)\, du$
    1. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 2 \left(1 + \frac{1}{u^{2}}\right)^{2}\right)\, du = - 2 \int \left(1 + \frac{1}{u^{2}}\right)^{2}\, du$
      
      There are multiple ways to do this integral.
      
      Method #1
      
      Rewrite the integrand:
      
      $\left(1 + \frac{1}{u^{2}}\right)^{2} = 1 + \frac{2}{u^{2}} + \frac{1}{u^{4}}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2}{u^{2}}\, du = 2 \int \frac{1}{u^{2}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      So, the result is: $- \frac{2}{u}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$
      
      The result is: $u - \frac{2}{u} - \frac{1}{3 u^{3}}$
      
      Method #2
      
      Rewrite the integrand:
      
      $\left(1 + \frac{1}{u^{2}}\right)^{2} = \frac{u^{4} + 2 u^{2} + 1}{u^{4}}$
      
      Rewrite the integrand:
      
      $\frac{u^{4} + 2 u^{2} + 1}{u^{4}} = 1 + \frac{2}{u^{2}} + \frac{1}{u^{4}}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2}{u^{2}}\, du = 2 \int \frac{1}{u^{2}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      So, the result is: $- \frac{2}{u}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$
      
      The result is: $u - \frac{2}{u} - \frac{1}{3 u^{3}}$
      
      So, the result is: $- 2 u + \frac{4}{u} + \frac{2}{3 u^{3}}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 2\, du = 2 u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2}{u^{2}}\, du = 2 \int \frac{1}{u^{2}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      
      So, the result is: $- \frac{2}{u}$
      
      The result is: $\frac{2}{u} + \frac{2}{3 u^{3}}$
    Now substitute $u$ back in:
    
    $\frac{2 \left(x - 1\right)^{\frac{3}{2}}}{3} + 2 \sqrt{x - 1}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{2}{\sqrt{x - 1}}\right)\, dx = - 2 \int \frac{1}{\sqrt{x - 1}}\, dx$
    1. Let $u = x - 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{\sqrt{u}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}$
      
      Now substitute $u$ back in:
      
      $2 \sqrt{x - 1}$
    So, the result is: $- 4 \sqrt{x - 1}$
  The result is: $\frac{2 \left(x - 1\right)^{\frac{3}{2}}}{3} - 2 \sqrt{x - 1}$
Now simplify:

$\frac{2 \left(x - 4\right) \sqrt{x - 1}}{3}$
Add the constant of integration:

$\frac{2 \left(x - 4\right) \sqrt{x - 1}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \left(x - 4\right) \sqrt{x - 1}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             
 |                                           3/2
 |   x - 2                _______   2*(x - 1)   
 | --------- dx = C - 2*\/ x - 1  + ------------
 |   _______                             3      
 | \/ x - 1                                     
 |                                              
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-4/3

$$- \frac{4}{3}$$

-4/3

$$- \frac{4}{3}$$

-4/3

Numerical answer [src]

-1.33333333280286

-1.33333333280286

The graph

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #1

Method #2