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Integral of xdx/√(x^2-3)^8
Identical expressions
xdx/√(x^ two - three)^ eight
xdx divide by √(x squared minus 3) to the power of 8
xdx divide by √(x to the power of two minus three) to the power of eight
xdx/√(x2-3)8
xdx/√x2-38
xdx/√(x²-3)⁸
xdx/√(x to the power of 2-3) to the power of 8
xdx/√x^2-3^8
xdx divide by √(x^2-3)^8
Similar expressions
xdx/√(x^2+3)^8

Solving integrals/ xdx/√(x^2-3)^8

Integral of xdx/√(x^2-3)^8 dx

The solution

You have entered [src]

 oo                
  /                
 |                 
 |       x         
 |  ------------ dx
 |             8   
 |     ________    
 |    /  2         
 |  \/  x  - 3     
 |                 
/                  
2

$$\int\limits_{2}^{\infty} \frac{x}{\left(\sqrt{x^{2} - 3}\right)^{8}}\, dx$$

Integral(x/(sqrt(x^2 - 3))^8, (x, 2, oo))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x}{\left(\sqrt{x^{2} - 3}\right)^{8}} = \frac{x}{x^{8} - 12 x^{6} + 54 x^{4} - 108 x^{2} + 81}$
2. Let $u = x^{2}$ .
  
  Then let $du = 2 x dx$ and substitute $du$ :
  
  $\int \frac{1}{2 u^{4} - 24 u^{3} + 108 u^{2} - 216 u + 162}\, du$
  1. Rewrite the integrand:
    
    $\frac{1}{2 u^{4} - 24 u^{3} + 108 u^{2} - 216 u + 162} = \frac{1}{2 \left(u - 3\right)^{4}}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 \left(u - 3\right)^{4}}\, du = \frac{\int \frac{1}{\left(u - 3\right)^{4}}\, du}{2}$
    1. Let $u = u - 3$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u^{4}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{3 \left(u - 3\right)^{3}}$
    So, the result is: $- \frac{1}{6 \left(u - 3\right)^{3}}$
  Now substitute $u$ back in:
  
  $- \frac{1}{6 \left(x^{2} - 3\right)^{3}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x}{\left(\sqrt{x^{2} - 3}\right)^{8}} = \frac{x}{x^{8} - 12 x^{6} + 54 x^{4} - 108 x^{2} + 81}$
2. Let $u = x^{2}$ .
  
  Then let $du = 2 x dx$ and substitute $du$ :
  
  $\int \frac{1}{2 u^{4} - 24 u^{3} + 108 u^{2} - 216 u + 162}\, du$
  1. Rewrite the integrand:
    
    $\frac{1}{2 u^{4} - 24 u^{3} + 108 u^{2} - 216 u + 162} = \frac{1}{2 \left(u - 3\right)^{4}}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 \left(u - 3\right)^{4}}\, du = \frac{\int \frac{1}{\left(u - 3\right)^{4}}\, du}{2}$
    1. Let $u = u - 3$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u^{4}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{3 \left(u - 3\right)^{3}}$
    So, the result is: $- \frac{1}{6 \left(u - 3\right)^{3}}$
  Now substitute $u$ back in:
  
  $- \frac{1}{6 \left(x^{2} - 3\right)^{3}}$
Add the constant of integration:

$- \frac{1}{6 \left(x^{2} - 3\right)^{3}}+ \mathrm{constant}$

The answer is:

$- \frac{1}{6 \left(x^{2} - 3\right)^{3}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                                   
 |      x                     1      
 | ------------ dx = C - ------------
 |            8                     3
 |    ________             /      2\ 
 |   /  2                6*\-3 + x / 
 | \/  x  - 3                        
 |                                   
/

$$\int \frac{x}{\left(\sqrt{x^{2} - 3}\right)^{8}}\, dx = C - \frac{1}{6 \left(x^{2} - 3\right)^{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/6

$$\frac{1}{6}$$

1/6

$$\frac{1}{6}$$

1/6

The graph

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

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Identical expressions

Similar expressions

Integral of xdx/√(x^2-3)^8 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2