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x/(x^(2)+x+1)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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  1              
  /              
 |               
 |      x        
 |  ---------- dx
 |   2           
 |  x  + x + 1   
 |               
/                
0                
$$\int\limits_{0}^{1} \frac{x}{x^{2} + x + 1}\, dx$$
Integral(x/(x^2 + x + 1), (x, 0, 1))
Detail solution
We have the integral:
  /               
 |                
 |       x        
 | 1*---------- dx
 |    2           
 |   x  + x + 1   
 |                
/                 
Rewrite the integrand
             /  1*2*x + 1   \                            
             |--------------|            / -1  \         
             |   2          |            |-----|         
    x        \1*x  + 1*x + 1/            \2*3/4/         
---------- = ---------------- + -------------------------
 2                  2                               2    
x  + x + 1                      /     ___       ___\     
                                |-2*\/ 3      \/ 3 |     
                                |--------*x - -----|  + 1
                                \   3           3  /     
or
  /                 
 |                  
 |       x          
 | 1*---------- dx  
 |    2            =
 |   x  + x + 1     
 |                  
/                   
  
                           /                            
                          |                             
                          |             1               
  /                    2* | ------------------------- dx
 |                        |                     2       
 |   1*2*x + 1            | /     ___       ___\        
 | -------------- dx      | |-2*\/ 3      \/ 3 |        
 |    2                   | |--------*x - -----|  + 1   
 | 1*x  + 1*x + 1         | \   3           3  /        
 |                        |                             
/                        /                              
-------------------- - ---------------------------------
         2                             3                
In the integral
  /                 
 |                  
 |   1*2*x + 1      
 | -------------- dx
 |    2             
 | 1*x  + 1*x + 1   
 |                  
/                   
--------------------
         2          
do replacement
         2
u = x + x 
then
the integral =
  /                     
 |                      
 |   1                  
 | ----- du             
 | 1 + u                
 |                      
/             log(1 + u)
----------- = ----------
     2            2     
do backward replacement
  /                                   
 |                                    
 |   1*2*x + 1                        
 | -------------- dx                  
 |    2                               
 | 1*x  + 1*x + 1                     
 |                        /         2\
/                      log\1 + x + x /
-------------------- = ---------------
         2                    2       
In the integral
     /                            
    |                             
    |             1               
-2* | ------------------------- dx
    |                     2       
    | /     ___       ___\        
    | |-2*\/ 3      \/ 3 |        
    | |--------*x - -----|  + 1   
    | \   3           3  /        
    |                             
   /                              
----------------------------------
                3                 
do replacement
        ___         ___
      \/ 3    2*x*\/ 3 
v = - ----- - ---------
        3         3    
then
the integral =
     /                      
    |                       
    |   1                   
-2* | ------ dv             
    |      2                
    | 1 + v                 
    |                       
   /              -2*atan(v)
--------------- = ----------
       3              3     
do backward replacement
     /                                                              
    |                                                               
    |             1                                                 
-2* | ------------------------- dx                                  
    |                     2                                         
    | /     ___       ___\                                          
    | |-2*\/ 3      \/ 3 |                                          
    | |--------*x - -----|  + 1                 /  ___         ___\ 
    | \   3           3  /              ___     |\/ 3    2*x*\/ 3 | 
    |                                -\/ 3 *atan|----- + ---------| 
   /                                            \  3         3    / 
---------------------------------- = -------------------------------
                3                                   3               
Solution is:
                                /  ___         ___\
                        ___     |\/ 3    2*x*\/ 3 |
       /         2\   \/ 3 *atan|----- + ---------|
    log\1 + x + x /             \  3         3    /
C + --------------- - -----------------------------
           2                        3              
The answer (Indefinite) [src]
                                                   /    ___          \
  /                                        ___     |2*\/ 3 *(1/2 + x)|
 |                        /         2\   \/ 3 *atan|-----------------|
 |     x               log\1 + x + x /             \        3        /
 | ---------- dx = C + --------------- - -----------------------------
 |  2                         2                        3              
 | x  + x + 1                                                         
 |                                                                    
/                                                                     
$${{\log \left(x^2+x+1\right)}\over{2}}-{{\arctan \left({{2\,x+1 }\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}$$
The graph
The answer [src]
              ___
log(3)   pi*\/ 3 
------ - --------
  2         18   
$${{\log 3}\over{2}}-{{\pi}\over{2\,3^{{{3}\over{2}}}}}$$
=
=
              ___
log(3)   pi*\/ 3 
------ - --------
  2         18   
$$- \frac{\sqrt{3} \pi}{18} + \frac{\log{\left(3 \right)}}{2}$$
Numerical answer [src]
0.247006250295019
0.247006250295019
The graph
Integral of x/(x^(2)+x+1) dx

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