Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of (-1+u)/(1+u^2)
Integral of cos2xsinx
Integral of √(1-cos(x))
Integral of 1/(1+x^(1/2))
Sum of series:
x^2/(x^3+1)
Derivative of:
x^2/(x^3+1)
Identical expressions
x^ two /(x^ three + one)
x squared divide by (x cubed plus 1)
x to the power of two divide by (x to the power of three plus one)
x2/(x3+1)
x2/x3+1
x²/(x³+1)
x to the power of 2/(x to the power of 3+1)
x^2/x^3+1
x^2 divide by (x^3+1)
x^2/(x^3+1)dx
Similar expressions
x^2/(x^3-1)
x^2/x^3+1dx

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

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

The solution

You have entered [src]

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

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

Integral(x^2/(x^3 + 1), (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: $\frac{\log{\left(x + 1 \right)}}{3} + \frac{\log{\left(x^{2} - x + 1 \right)}}{3}$
Now simplify:

$\frac{\log{\left(x^{3} + 1 \right)}}{3}$
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\x  + 1/
 | ------ dx = C + -----------
 |  3                   3     
 | x  + 1                     
 |                            
/

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

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.

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2