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1/sqrt^3(x+2)

Integral of 1/sqrt^3(x-2) dx

The solution

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  1              
  /              
 |               
 |      1        
 |  ---------- dx
 |           3   
 |    _______    
 |  \/ x - 2     
 |               
/                
-oo

$$\int\limits_{-\infty}^{1} \frac{1}{\left(\sqrt{x - 2}\right)^{3}}\, dx$$

Integral(1/((sqrt(x - 2))^3), (x, -oo, 1))

Detail solution

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

$- \frac{2}{\sqrt{x - 2}}+ \mathrm{constant}$

The answer is:

$- \frac{2}{\sqrt{x - 2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              
 |                               
 |     1                   2     
 | ---------- dx = C - ----------
 |          3            ________
 |   _______           \/ -2 + x 
 | \/ x - 2                      
 |                               
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

2*I

$$2 i$$

2*I

$$2 i$$

2*i

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

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Identical expressions

Similar expressions

Integral of 1/sqrt^3(x-2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2