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Integral of d{x}:
Integral of x^3*exp(x^2)
Integral of x^2*a^x
Integral of sec³x
Integral of x*3^x
Graphing y =:
(x^3)/(4-x^2)
Derivative of:
(x^3)/(4-x^2)
Identical expressions
(x^ three)/(four -x^ two)
(x cubed ) divide by (4 minus x squared )
(x to the power of three) divide by (four minus x to the power of two)
(x3)/(4-x2)
x3/4-x2
(x³)/(4-x²)
(x to the power of 3)/(4-x to the power of 2)
x^3/4-x^2
(x^3) divide by (4-x^2)
(x^3)/(4-x^2)dx
Similar expressions
(x^3)/(4+x^2)

Solving integrals/ (x^3)/(4-x^2)

Integral of (x^3)/(4-x^2) dx

The solution

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  1          
  /          
 |           
 |     3     
 |    x      
 |  ------ dx
 |       2   
 |  4 - x    
 |           
/            
0

$$\int\limits_{0}^{1} \frac{x^{3}}{4 - x^{2}}\, dx$$

Integral(x^3/(4 - x^2), (x, 0, 1))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-1/2 - 2*log(3) + 2*log(4)

$$- 2 \log{\left(3 \right)} - \frac{1}{2} + 2 \log{\left(4 \right)}$$

-1/2 - 2*log(3) + 2*log(4)

$$- 2 \log{\left(3 \right)} - \frac{1}{2} + 2 \log{\left(4 \right)}$$

-1/2 - 2*log(3) + 2*log(4)

Numerical answer [src]

0.0753641449035619

0.0753641449035619

The graph

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

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Identical expressions

Similar expressions

Integral of (x^3)/(4-x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3