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

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

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

The solution

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

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

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

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3 x^{2}}{x^{3} + 1}\, dx = 3 \int \frac{x^{2}}{x^{3} + 1}\, dx$
  1. 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 $\frac{du}{3}$ :
      
      $\int \frac{1}{9 u}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{3 u}\, du = \frac{\int \frac{1}{u}\, du}{3}$
      
      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:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2 x - 1}{3 \left(x^{2} - x + 1\right)}\, dx = \frac{\int \frac{2 x - 1}{x^{2} - x + 1}\, dx}{3}$
      
      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: $\frac{\log{\left(x^{2} - x + 1 \right)}}{3}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{3 \left(x + 1\right)}\, dx = \frac{\int \frac{1}{x + 1}\, dx}{3}$
      
      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: $\frac{\log{\left(x + 1 \right)}}{3} + \frac{\log{\left(x^{2} - x + 1 \right)}}{3}$
  So, the result is: $\log{\left(x^{3} + 1 \right)}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 2\, dx = 2 x$
The result is: $2 x + \log{\left(x^{3} + 1 \right)}$
Now simplify:

$2 x + \log{\left(x^{3} + 1 \right)}$
Add the constant of integration:

$2 x + \log{\left(x^{3} + 1 \right)}+ \mathrm{constant}$

The answer is:

$2 x + \log{\left(x^{3} + 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\log \left(x^3+1\right)+2\,x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

2 + log(2)

$$\log 2+2$$

2 + log(2)

$$\log{\left(2 \right)} + 2$$

Simplify

Numerical answer [src]

2.69314718055995

2.69314718055995

The graph

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2