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Integral of d{x}:
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Integral of arctan(x)
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Integral of x-1
Graphing y =:
x^3/(2+x^2)
Limit of the function:
x^3/(2+x^2)
Identical expressions
x^ three /(two +x^ two)
x cubed divide by (2 plus x squared )
x to the power of three divide by (two plus x to the power of two)
x3/(2+x2)
x3/2+x2
x³/(2+x²)
x to the power of 3/(2+x to the power of 2)
x^3/2+x^2
x^3 divide by (2+x^2)
x^3/(2+x^2)dx
Similar expressions
x^3/(2-x^2)

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

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

The solution

You have entered [src]

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

0.0945348918918356

0.0945348918918356

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/(2+x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2