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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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  1                
  /                
 |                 
 |       x         
 |  ------------ dx
 |   2             
 |  x  - 3*x + 3   
 |                 
/                  
0                  
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - 3 x\right) + 3}\, dx$$
Integral(x/(x^2 - 3*x + 3), (x, 0, 1))
Detail solution
We have the integral:
  /               
 |                
 |      x         
 | ------------ dx
 |  2             
 | x  - 3*x + 3   
 |                
/                 
Rewrite the integrand
               /  2*x - 3   \                            
               |------------|            /  3  \         
               | 2          |            |-----|         
     x         \x  - 3*x + 3/            \2*3/4/         
------------ = -------------- + -------------------------
 2                   2                              2    
x  - 3*x + 3                    /     ___          \     
                                |-2*\/ 3        ___|     
                                |--------*x + \/ 3 |  + 1
                                \   3              /     
or
  /                 
 |                  
 |      x           
 | ------------ dx  
 |  2              =
 | x  - 3*x + 3     
 |                  
/                   
  
  /                                                   
 |                                                    
 |   2*x - 3                                          
 | ------------ dx                                    
 |  2                                                 
 | x  - 3*x + 3          /                            
 |                      |                             
/                       |             1               
------------------ + 2* | ------------------------- dx
        2               |                     2       
                        | /     ___          \        
                        | |-2*\/ 3        ___|        
                        | |--------*x + \/ 3 |  + 1   
                        | \   3              /        
                        |                             
                       /                              
In the integral
  /               
 |                
 |   2*x - 3      
 | ------------ dx
 |  2             
 | x  - 3*x + 3   
 |                
/                 
------------------
        2         
do replacement
     2      
u = x  - 3*x
then
the integral =
  /                     
 |                      
 |   1                  
 | ----- du             
 | 3 + u                
 |                      
/             log(3 + u)
----------- = ----------
     2            2     
do backward replacement
  /                                   
 |                                    
 |   2*x - 3                          
 | ------------ dx                    
 |  2                                 
 | x  - 3*x + 3                       
 |                      /     2      \
/                    log\3 + x  - 3*x/
------------------ = -----------------
        2                    2        
In the integral
    /                            
   |                             
   |             1               
2* | ------------------------- dx
   |                     2       
   | /     ___          \        
   | |-2*\/ 3        ___|        
   | |--------*x + \/ 3 |  + 1   
   | \   3              /        
   |                             
  /                              
do replacement
                  ___
      ___   2*x*\/ 3 
v = \/ 3  - ---------
                3    
then
the integral =
    /                     
   |                      
   |   1                  
2* | ------ dv = 2*atan(v)
   |      2               
   | 1 + v                
   |                      
  /                       
do backward replacement
    /                                                              
   |                                          /                ___\
   |             1                    ___     |    ___   2*x*\/ 3 |
2* | ------------------------- dx = \/ 3 *atan|- \/ 3  + ---------|
   |                     2                    \              3    /
   | /     ___          \                                          
   | |-2*\/ 3        ___|                                          
   | |--------*x + \/ 3 |  + 1                                     
   | \   3              /                                          
   |                                                               
  /                                                                
Solution is:
       /     2      \             /                ___\
    log\3 + x  - 3*x/     ___     |    ___   2*x*\/ 3 |
C + ----------------- + \/ 3 *atan|- \/ 3  + ---------|
            2                     \              3    /
The answer (Indefinite) [src]
  /                                                                        
 |                          /     2      \             /    ___           \
 |      x                log\3 + x  - 3*x/     ___     |2*\/ 3 *(-3/2 + x)|
 | ------------ dx = C + ----------------- + \/ 3 *atan|------------------|
 |  2                            2                     \        3         /
 | x  - 3*x + 3                                                            
 |                                                                         
/                                                                          
$$\int \frac{x}{\left(x^{2} - 3 x\right) + 3}\, dx = C + \frac{\log{\left(x^{2} - 3 x + 3 \right)}}{2} + \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x - \frac{3}{2}\right)}{3} \right)}$$
The graph
The answer [src]
                ___
  log(3)   pi*\/ 3 
- ------ + --------
    2         6    
$$- \frac{\log{\left(3 \right)}}{2} + \frac{\sqrt{3} \pi}{6}$$
=
=
                ___
  log(3)   pi*\/ 3 
- ------ + --------
    2         6    
$$- \frac{\log{\left(3 \right)}}{2} + \frac{\sqrt{3} \pi}{6}$$
-log(3)/2 + pi*sqrt(3)/6
Numerical answer [src]
0.357593537783054
0.357593537783054
The graph
Integral of x/(x^2-3x+3) dx

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