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Integral of d{x}:
Integral of (x+4)/(x^2(x-1))
Integral of x^4/(9-x^2)
Integral of (x^4-53sqrtx^2+1)
Integral of x^4-3x^2+2-----x^4+x^2
Limit of the function:
x^4/(9-x^2)
Identical expressions
x^ four /(nine -x^ two)
x to the power of 4 divide by (9 minus x squared )
x to the power of four divide by (nine minus x to the power of two)
x4/(9-x2)
x4/9-x2
x⁴/(9-x²)
x to the power of 4/(9-x to the power of 2)
x^4/9-x^2
x^4 divide by (9-x^2)
x^4/(9-x^2)dx
Similar expressions
x^4/(9+x^2)
What you mean?
x^4/(9 - x^2)
x^4*2/(9 - x)
x^4/((9 - x)^2)
x^4/9 - x^2
x*4/(9 - x^2)
x*4*2/(9 - x)
x^4/9 - x^2

You entered:

x^4/(9-x^2)

What you mean?

x^4/(9 - x^2)

x^4*2/(9 - x)

x^4/((9 - x)^2)

x^4/9 - x^2

x*4/(9 - x^2)

x*4*2/(9 - x)

x^4/9 - x^2

Integral of x^4/(9-x^2) dx

The solution

You have entered [src]

  1          
  /          
 |           
 |     4     
 |    x      
 |  ------ dx
 |       2   
 |  9 - x    
 |           
/            
0

$$\int\limits_{0}^{1} \frac{x^{4}}{9 - x^{2}}\, dx$$

Integral(x^4/(9 - x^2), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x^{4}}{9 - x^{2}} = - x^{2} - 9 + \frac{27}{2 \left(x + 3\right)} - \frac{27}{2 \left(x - 3\right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- x^{2}\right)\, dx = - \int x^{2}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    So, the result is: $- \frac{x^{3}}{3}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(-9\right)\, dx = - 9 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{27}{2 \left(x + 3\right)}\, dx = \frac{27 \int \frac{1}{x + 3}\, dx}{2}$
    1. Let $u = x + 3$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 3 \right)}$
    So, the result is: $\frac{27 \log{\left(x + 3 \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{27}{2 \left(x - 3\right)}\right)\, dx = - \frac{27 \int \frac{1}{x - 3}\, dx}{2}$
    1. Let $u = x - 3$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x - 3 \right)}$
    So, the result is: $- \frac{27 \log{\left(x - 3 \right)}}{2}$
  The result is: $- \frac{x^{3}}{3} - 9 x - \frac{27 \log{\left(x - 3 \right)}}{2} + \frac{27 \log{\left(x + 3 \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x^{4}}{9 - x^{2}} = - \frac{x^{4}}{x^{2} - 9}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{x^{4}}{x^{2} - 9}\right)\, dx = - \int \frac{x^{4}}{x^{2} - 9}\, dx$
  1. Rewrite the integrand:
    
    $\frac{x^{4}}{x^{2} - 9} = x^{2} + 9 - \frac{27}{2 \left(x + 3\right)} + \frac{27}{2 \left(x - 3\right)}$
  2. Integrate term-by-term:
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 9\, dx = 9 x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{27}{2 \left(x + 3\right)}\right)\, dx = - \frac{27 \int \frac{1}{x + 3}\, dx}{2}$
      
      Let $u = x + 3$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 3 \right)}$
      
      So, the result is: $- \frac{27 \log{\left(x + 3 \right)}}{2}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{27}{2 \left(x - 3\right)}\, dx = \frac{27 \int \frac{1}{x - 3}\, dx}{2}$
      
      Let $u = x - 3$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x - 3 \right)}$
      
      So, the result is: $\frac{27 \log{\left(x - 3 \right)}}{2}$
    The result is: $\frac{x^{3}}{3} + 9 x + \frac{27 \log{\left(x - 3 \right)}}{2} - \frac{27 \log{\left(x + 3 \right)}}{2}$
  So, the result is: $- \frac{x^{3}}{3} - 9 x - \frac{27 \log{\left(x - 3 \right)}}{2} + \frac{27 \log{\left(x + 3 \right)}}{2}$
Add the constant of integration:

$- \frac{x^{3}}{3} - 9 x - \frac{27 \log{\left(x - 3 \right)}}{2} + \frac{27 \log{\left(x + 3 \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{x^{3}}{3} - 9 x - \frac{27 \log{\left(x - 3 \right)}}{2} + \frac{27 \log{\left(x + 3 \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                         
 |                                                          
 |    4                                    3                
 |   x                   27*log(-3 + x)   x    27*log(3 + x)
 | ------ dx = C - 9*x - -------------- - -- + -------------
 |      2                      2          3          2      
 | 9 - x                                                    
 |                                                          
/

$${{27\,\log \left(x+3\right)}\over{2}}-{{x^3+27\,x}\over{3}}-{{27\, \log \left(x-3\right)}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  28   27*log(2)   27*log(4)
- -- - --------- + ---------
  3        2           2

$${{81\,\log 4-81\,\log 2-56}\over{6}}$$

  28   27*log(2)   27*log(4)
- -- - --------- + ---------
  3        2           2

$$- \frac{27 \log{\left(2 \right)}}{2} - \frac{28}{3} + \frac{27 \log{\left(4 \right)}}{2}$$

Упростить

Numerical answer [src]

0.0241536042259283

0.0241536042259283

The graph

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

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Identical expressions

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What you mean?

Integral of x^4/(9-x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2