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Integral of xdx/(x^2+2)^5
Identical expressions
xdx/(x^ two + two)^ five
xdx divide by (x squared plus 2) to the power of 5
xdx divide by (x to the power of two plus two) to the power of five
xdx/(x2+2)5
xdx/x2+25
xdx/(x²+2)⁵
xdx/(x to the power of 2+2) to the power of 5
xdx/x^2+2^5
xdx divide by (x^2+2)^5
Similar expressions
xdx/(x^2-2)^5

Solving integrals/ xdx/(x^2+2)^5

Integral of xdx/(x^2+2)^5 dx

The solution

You have entered [src]

  1             
  /             
 |              
 |      x       
 |  --------- dx
 |          5   
 |  / 2    \    
 |  \x  + 2/    
 |              
/               
0

$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} + 2\right)^{5}}\, dx$$

Integral(x/(x^2 + 2)^5, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x}{\left(x^{2} + 2\right)^{5}} = \frac{x}{x^{10} + 10 x^{8} + 40 x^{6} + 80 x^{4} + 80 x^{2} + 32}$
2. Let $u = x^{2}$ .
  
  Then let $du = 2 x dx$ and substitute $du$ :
  
  $\int \frac{1}{2 u^{5} + 20 u^{4} + 80 u^{3} + 160 u^{2} + 160 u + 64}\, du$
  1. Rewrite the integrand:
    
    $\frac{1}{2 u^{5} + 20 u^{4} + 80 u^{3} + 160 u^{2} + 160 u + 64} = \frac{1}{2 \left(u + 2\right)^{5}}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 \left(u + 2\right)^{5}}\, du = \frac{\int \frac{1}{\left(u + 2\right)^{5}}\, du}{2}$
    1. Let $u = u + 2$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u^{5}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{5}}\, du = - \frac{1}{4 u^{4}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{4 \left(u + 2\right)^{4}}$
    So, the result is: $- \frac{1}{8 \left(u + 2\right)^{4}}$
  Now substitute $u$ back in:
  
  $- \frac{1}{8 \left(x^{2} + 2\right)^{4}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x}{\left(x^{2} + 2\right)^{5}} = \frac{x}{x^{10} + 10 x^{8} + 40 x^{6} + 80 x^{4} + 80 x^{2} + 32}$
2. Let $u = x^{2}$ .
  
  Then let $du = 2 x dx$ and substitute $du$ :
  
  $\int \frac{1}{2 u^{5} + 20 u^{4} + 80 u^{3} + 160 u^{2} + 160 u + 64}\, du$
  1. Rewrite the integrand:
    
    $\frac{1}{2 u^{5} + 20 u^{4} + 80 u^{3} + 160 u^{2} + 160 u + 64} = \frac{1}{2 \left(u + 2\right)^{5}}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 \left(u + 2\right)^{5}}\, du = \frac{\int \frac{1}{\left(u + 2\right)^{5}}\, du}{2}$
    1. Let $u = u + 2$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u^{5}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{5}}\, du = - \frac{1}{4 u^{4}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{4 \left(u + 2\right)^{4}}$
    So, the result is: $- \frac{1}{8 \left(u + 2\right)^{4}}$
  Now substitute $u$ back in:
  
  $- \frac{1}{8 \left(x^{2} + 2\right)^{4}}$
Add the constant of integration:

$- \frac{1}{8 \left(x^{2} + 2\right)^{4}}+ \mathrm{constant}$

The answer is:

$- \frac{1}{8 \left(x^{2} + 2\right)^{4}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              
 |                               
 |     x                   1     
 | --------- dx = C - -----------
 |         5                    4
 | / 2    \             /     2\ 
 | \x  + 2/           8*\2 + x / 
 |                               
/

$$\int \frac{x}{\left(x^{2} + 2\right)^{5}}\, dx = C - \frac{1}{8 \left(x^{2} + 2\right)^{4}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  65 
-----
10368

$$\frac{65}{10368}$$

  65 
-----
10368

$$\frac{65}{10368}$$

65/10368

Numerical answer [src]

0.00626929012345679

0.00626929012345679

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+2)^5 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2