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Integral of d{x}:
Integral of x^2/x
Integral of 6/x^2
Integral of 2/sinx
Integral of xln(2-x)
Identical expressions
xdx/(x^ two + four)^ six
xdx divide by (x squared plus 4) to the power of 6
xdx divide by (x to the power of two plus four) to the power of six
xdx/(x2+4)6
xdx/x2+46
xdx/(x²+4)⁶
xdx/(x to the power of 2+4) to the power of 6
xdx/x^2+4^6
xdx divide by (x^2+4)^6
Similar expressions
xdx/(x^2-4)^6

Solving integrals/ xdx/(x^2+4)^6

Integral of xdx/(x^2+4)^6 dx

The solution

You have entered [src]

  1             
  /             
 |              
 |      x       
 |  --------- dx
 |          6   
 |  / 2    \    
 |  \x  + 4/    
 |              
/               
0

\int\limits_{0}^{1} \frac{x}{\left(x^{2} + 4\right)^{6}}\, dx

Integral(x/(x^2 + 4)^6, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x}{\left(x^{2} + 4\right)^{6}} = \frac{x}{x^{12} + 24 x^{10} + 240 x^{8} + 1280 x^{6} + 3840 x^{4} + 6144 x^{2} + 4096}$
2. Let $u = x^{2}$ .
  
  Then let $du = 2 x dx$ and substitute $du$ :
  
  $\int \frac{1}{2 u^{6} + 48 u^{5} + 480 u^{4} + 2560 u^{3} + 7680 u^{2} + 12288 u + 8192}\, du$
  1. Rewrite the integrand:
    
    $\frac{1}{2 u^{6} + 48 u^{5} + 480 u^{4} + 2560 u^{3} + 7680 u^{2} + 12288 u + 8192} = \frac{1}{2 \left(u + 4\right)^{6}}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 \left(u + 4\right)^{6}}\, du = \frac{\int \frac{1}{\left(u + 4\right)^{6}}\, du}{2}$
    1. Let $u = u + 4$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u^{6}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{6}}\, du = - \frac{1}{5 u^{5}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{5 \left(u + 4\right)^{5}}$
    So, the result is: $- \frac{1}{10 \left(u + 4\right)^{5}}$
  Now substitute $u$ back in:
  
  $- \frac{1}{10 \left(x^{2} + 4\right)^{5}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x}{\left(x^{2} + 4\right)^{6}} = \frac{x}{x^{12} + 24 x^{10} + 240 x^{8} + 1280 x^{6} + 3840 x^{4} + 6144 x^{2} + 4096}$
2. Let $u = x^{2}$ .
  
  Then let $du = 2 x dx$ and substitute $du$ :
  
  $\int \frac{1}{2 u^{6} + 48 u^{5} + 480 u^{4} + 2560 u^{3} + 7680 u^{2} + 12288 u + 8192}\, du$
  1. Rewrite the integrand:
    
    $\frac{1}{2 u^{6} + 48 u^{5} + 480 u^{4} + 2560 u^{3} + 7680 u^{2} + 12288 u + 8192} = \frac{1}{2 \left(u + 4\right)^{6}}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 \left(u + 4\right)^{6}}\, du = \frac{\int \frac{1}{\left(u + 4\right)^{6}}\, du}{2}$
    1. Let $u = u + 4$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u^{6}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{6}}\, du = - \frac{1}{5 u^{5}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{5 \left(u + 4\right)^{5}}$
    So, the result is: $- \frac{1}{10 \left(u + 4\right)^{5}}$
  Now substitute $u$ back in:
  
  $- \frac{1}{10 \left(x^{2} + 4\right)^{5}}$
Add the constant of integration:

$- \frac{1}{10 \left(x^{2} + 4\right)^{5}}+ \mathrm{constant}$

The answer is:

- \frac{1}{10 \left(x^{2} + 4\right)^{5}}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                               
 |                                
 |     x                   1      
 | --------- dx = C - ------------
 |         6                     5
 | / 2    \              /     2\ 
 | \x  + 4/           10*\4 + x / 
 |                                
/

\int \frac{x}{\left(x^{2} + 4\right)^{6}}\, dx = C - \frac{1}{10 \left(x^{2} + 4\right)^{5}}

Simplify

The graph

Plot the graph f(x)

The answer [src]

  2101  
--------
32000000

\frac{2101}{32000000}

  2101  
--------
32000000

\frac{2101}{32000000}

2101/32000000

Numerical answer [src]

6.565625e-5

6.565625e-5

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of xdx/(x^2+4)^6 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2