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8x^ two /(x^ three + two)^ three
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8x^2/(x^3+2)^3dx
Similar expressions
8x^2/(x^3-2)^3

Solving integrals/ 8x^2/(x^3+2)^3

Integral of 8x^2/(x^3+2)^3 dx

The solution

You have entered [src]

  1             
  /             
 |              
 |        2     
 |     8*x      
 |  --------- dx
 |          3   
 |  / 3    \    
 |  \x  + 2/    
 |              
/               
0

$$\int\limits_{0}^{1} \frac{8 x^{2}}{\left(x^{3} + 2\right)^{3}}\, dx$$

Integral((8*x^2)/(x^3 + 2)^3, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{8 x^{2}}{\left(x^{3} + 2\right)^{3}} = \frac{8 x^{2}}{x^{9} + 6 x^{6} + 12 x^{3} + 8}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{8 x^{2}}{x^{9} + 6 x^{6} + 12 x^{3} + 8}\, dx = 8 \int \frac{x^{2}}{x^{9} + 6 x^{6} + 12 x^{3} + 8}\, dx$
  1. Let $u = x^{3}$ .
    
    Then let $du = 3 x^{2} dx$ and substitute $du$ :
    
    $\int \frac{1}{3 u^{3} + 18 u^{2} + 36 u + 24}\, du$
    1. Rewrite the integrand:
      
      $\frac{1}{3 u^{3} + 18 u^{2} + 36 u + 24} = \frac{1}{3 \left(u + 2\right)^{3}}$
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{3 \left(u + 2\right)^{3}}\, du = \frac{\int \frac{1}{\left(u + 2\right)^{3}}\, du}{3}$
      
      Let $u = u + 2$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u^{3}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{3}}\, du = - \frac{1}{2 u^{2}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{2 \left(u + 2\right)^{2}}$
      
      So, the result is: $- \frac{1}{6 \left(u + 2\right)^{2}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{6 \left(x^{3} + 2\right)^{2}}$
  So, the result is: $- \frac{4}{3 \left(x^{3} + 2\right)^{2}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{8 x^{2}}{\left(x^{3} + 2\right)^{3}} = \frac{8 x^{2}}{x^{9} + 6 x^{6} + 12 x^{3} + 8}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{8 x^{2}}{x^{9} + 6 x^{6} + 12 x^{3} + 8}\, dx = 8 \int \frac{x^{2}}{x^{9} + 6 x^{6} + 12 x^{3} + 8}\, dx$
  1. Let $u = x^{3}$ .
    
    Then let $du = 3 x^{2} dx$ and substitute $du$ :
    
    $\int \frac{1}{3 u^{3} + 18 u^{2} + 36 u + 24}\, du$
    1. Rewrite the integrand:
      
      $\frac{1}{3 u^{3} + 18 u^{2} + 36 u + 24} = \frac{1}{3 \left(u + 2\right)^{3}}$
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{3 \left(u + 2\right)^{3}}\, du = \frac{\int \frac{1}{\left(u + 2\right)^{3}}\, du}{3}$
      
      Let $u = u + 2$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u^{3}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{3}}\, du = - \frac{1}{2 u^{2}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{2 \left(u + 2\right)^{2}}$
      
      So, the result is: $- \frac{1}{6 \left(u + 2\right)^{2}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{6 \left(x^{3} + 2\right)^{2}}$
  So, the result is: $- \frac{4}{3 \left(x^{3} + 2\right)^{2}}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                              
 |                               
 |       2                       
 |    8*x                  4     
 | --------- dx = C - -----------
 |         3                    2
 | / 3    \             /     3\ 
 | \x  + 2/           3*\2 + x / 
 |                               
/

$$\int \frac{8 x^{2}}{\left(x^{3} + 2\right)^{3}}\, dx = C - \frac{4}{3 \left(x^{3} + 2\right)^{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

5/27

$$\frac{5}{27}$$

5/27

$$\frac{5}{27}$$

5/27

Numerical answer [src]

0.185185185185185

0.185185185185185

The graph

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

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Identical expressions

Similar expressions

Integral of 8x^2/(x^3+2)^3 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2