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(3x+2)/(x^2-x+1)

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

Limits of integration:

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The graph:

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Piecewise:

The solution

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  1              
  /              
 |               
 |   3*x + 2     
 |  ---------- dx
 |   2           
 |  x  - x + 1   
 |               
/                
0                
$$\int\limits_{0}^{1} \frac{3 x + 2}{x^{2} - x + 1}\, dx$$
Integral((3*x + 2)/(x^2 - x + 1), (x, 0, 1))
Detail solution
We have the integral:
  /               
 |                
 |    3*x + 2     
 | 1*---------- dx
 |    2           
 |   x  - x + 1   
 |                
/                 
Rewrite the integrand
                1*2*x - 1                              
             3*------------            /  7  \         
                  2                    |-----|         
 3*x + 2       1*x  - x + 1            \2*3/4/         
---------- = -------------- + -------------------------
 2                 2                              2    
x  - x + 1                    /     ___       ___\     
                              |-2*\/ 3      \/ 3 |     
                              |--------*x + -----|  + 1
                              \   3           3  /     
or
  /                 
 |                  
 |    3*x + 2       
 | 1*---------- dx  
 |    2            =
 |   x  - x + 1     
 |                  
/                   
  
                            /                            
                           |                             
                           |             1               
    /                  14* | ------------------------- dx
   |                       |                     2       
   |  1*2*x - 1            | /     ___       ___\        
3* | ------------ dx       | |-2*\/ 3      \/ 3 |        
   |    2                  | |--------*x + -----|  + 1   
   | 1*x  - x + 1          | \   3           3  /        
   |                       |                             
  /                       /                              
-------------------- + ----------------------------------
         2                             3                 
In the integral
    /               
   |                
   |  1*2*x - 1     
3* | ------------ dx
   |    2           
   | 1*x  - x + 1   
   |                
  /                 
--------------------
         2          
do replacement
     2    
u = x  - x
then
the integral =
    /                       
   |                        
   |   1                    
3* | ----- du               
   | 1 + u                  
   |                        
  /             3*log(1 + u)
------------- = ------------
      2              2      
do backward replacement
    /                                   
   |                                    
   |  1*2*x - 1                         
3* | ------------ dx                    
   |    2                               
   | 1*x  - x + 1                       
   |                        /     2    \
  /                    3*log\1 + x  - x/
-------------------- = -----------------
         2                     2        
In the integral
     /                            
    |                             
    |             1               
14* | ------------------------- dx
    |                     2       
    | /     ___       ___\        
    | |-2*\/ 3      \/ 3 |        
    | |--------*x + -----|  + 1   
    | \   3           3  /        
    |                             
   /                              
----------------------------------
                3                 
do replacement
      ___         ___
    \/ 3    2*x*\/ 3 
v = ----- - ---------
      3         3    
then
the integral =
     /                      
    |                       
    |   1                   
14* | ------ dv             
    |      2                
    | 1 + v                 
    |                       
   /              14*atan(v)
--------------- = ----------
       3              3     
do backward replacement
     /                                                                
    |                                                                 
    |             1                                                   
14* | ------------------------- dx                                    
    |                     2                                           
    | /     ___       ___\                                            
    | |-2*\/ 3      \/ 3 |                                            
    | |--------*x + -----|  + 1                  /    ___         ___\
    | \   3           3  /               ___     |  \/ 3    2*x*\/ 3 |
    |                                7*\/ 3 *atan|- ----- + ---------|
   /                                             \    3         3    /
---------------------------------- = ---------------------------------
                3                                    3                
Solution is:
                                    /    ___         ___\
                            ___     |  \/ 3    2*x*\/ 3 |
         /     2    \   7*\/ 3 *atan|- ----- + ---------|
    3*log\1 + x  - x/               \    3         3    /
C + ----------------- + ---------------------------------
            2                           3                
The answer (Indefinite) [src]
                                                       /    ___           \
  /                                            ___     |2*\/ 3 *(-1/2 + x)|
 |                          /     2    \   7*\/ 3 *atan|------------------|
 |  3*x + 2            3*log\1 + x  - x/               \        3         /
 | ---------- dx = C + ----------------- + --------------------------------
 |  2                          2                          3                
 | x  - x + 1                                                              
 |                                                                         
/                                                                          
$${{3\,\log \left(x^2-x+1\right)}\over{2}}+{{7\,\arctan \left({{2\,x- 1}\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}$$
The graph
The answer [src]
       ___
7*pi*\/ 3 
----------
    9     
$${{7\,\pi}\over{3^{{{3}\over{2}}}}}$$
=
=
       ___
7*pi*\/ 3 
----------
    9     
$$\frac{7 \sqrt{3} \pi}{9}$$
Numerical answer [src]
4.23219851654651
4.23219851654651
The graph
Integral of (3x+2)/(x^2-x+1) dx

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