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Derivative of:
x^2/(1+x^3)
Limit of the function:
x^2/(1+x^3)
Identical expressions
x^ two /(one +x^ three)
x squared divide by (1 plus x cubed )
x to the power of two divide by (one plus x to the power of three)
x2/(1+x3)
x2/1+x3
x²/(1+x³)
x to the power of 2/(1+x to the power of 3)
x^2/1+x^3
x^2 divide by (1+x^3)
x^2/(1+x^3)dx
Similar expressions
x^2/(1-x^3)

Solving integrals/ x^2/(1+x^3)

Integral of x^2/(1+x^3) dx

The solution

You have entered [src]

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

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

Integral(x^2/(1 + x^3), (x, 0, 1))

Detail solution

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

$\frac{\log{\left(x^{3} + 1 \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(x^{3} + 1 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                           
 |                            
 |    2               /     3\
 |   x             log\1 + x /
 | ------ dx = C + -----------
 |      3               3     
 | 1 + x                      
 |                            
/

$$\int \frac{x^{2}}{x^{3} + 1}\, dx = C + \frac{\log{\left(x^{3} + 1 \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(2)
------
  3

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

log(2)
------
  3

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

log(2)/3

Numerical answer [src]

0.231049060186648

0.231049060186648

The graph

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

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Identical expressions

Similar expressions

Integral of x^2/(1+x^3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2