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

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

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

You have entered [src]
  1                
  /                
 |                 
 |     x - 1       
 |  ------------ dx
 |   2             
 |  x  + 2*x + 3   
 |                 
/                  
0                  
$$\int\limits_{0}^{1} \frac{x - 1}{\left(x^{2} + 2 x\right) + 3}\, dx$$
Integral((x - 1)/(x^2 + 2*x + 3), (x, 0, 1))
Detail solution
We have the integral:
  /               
 |                
 |    x - 1       
 | ------------ dx
 |  2             
 | x  + 2*x + 3   
 |                
/                 
Rewrite the integrand
               /  2*x + 2   \                           
               |------------|            /-2 \          
               | 2          |            |---|          
   x - 1       \x  + 2*x + 3/            \ 2 /          
------------ = -------------- + ------------------------
 2                   2                             2    
x  + 2*x + 3                    /   ___        ___\     
                                |-\/ 2       \/ 2 |     
                                |-------*x - -----|  + 1
                                \   2          2  /     
or
  /                 
 |                  
 |    x - 1         
 | ------------ dx  
 |  2              =
 | x  + 2*x + 3     
 |                  
/                   
  
  /                                                
 |                                                 
 |   2*x + 2                                       
 | ------------ dx                                 
 |  2                                              
 | x  + 2*x + 3        /                           
 |                    |                            
/                     |            1               
------------------ -  | ------------------------ dx
        2             |                    2       
                      | /   ___        ___\        
                      | |-\/ 2       \/ 2 |        
                      | |-------*x - -----|  + 1   
                      | \   2          2  /        
                      |                            
                     /                             
In the integral
  /               
 |                
 |   2*x + 2      
 | ------------ dx
 |  2             
 | x  + 2*x + 3   
 |                
/                 
------------------
        2         
do replacement
     2      
u = x  + 2*x
then
the integral =
  /                     
 |                      
 |   1                  
 | ----- du             
 | 3 + u                
 |                      
/             log(3 + u)
----------- = ----------
     2            2     
do backward replacement
  /                                   
 |                                    
 |   2*x + 2                          
 | ------------ dx                    
 |  2                                 
 | x  + 2*x + 3                       
 |                      /     2      \
/                    log\3 + x  + 2*x/
------------------ = -----------------
        2                    2        
In the integral
   /                           
  |                            
  |            1               
- | ------------------------ dx
  |                    2       
  | /   ___        ___\        
  | |-\/ 2       \/ 2 |        
  | |-------*x - -----|  + 1   
  | \   2          2  /        
  |                            
 /                             
do replacement
        ___       ___
      \/ 2    x*\/ 2 
v = - ----- - -------
        2        2   
then
the integral =
   /                    
  |                     
  |   1                 
- | ------ dv = -atan(v)
  |      2              
  | 1 + v               
  |                     
 /                      
do backward replacement
   /                                                          
  |                                          /  ___       ___\
  |            1                     ___     |\/ 2    x*\/ 2 |
- | ------------------------ dx = -\/ 2 *atan|----- + -------|
  |                    2                     \  2        2   /
  | /   ___        ___\                                       
  | |-\/ 2       \/ 2 |                                       
  | |-------*x - -----|  + 1                                  
  | \   2          2  /                                       
  |                                                           
 /                                                            
Solution is:
       /     2      \             /  ___       ___\
    log\3 + x  + 2*x/     ___     |\/ 2    x*\/ 2 |
C + ----------------- - \/ 2 *atan|----- + -------|
            2                     \  2        2   /
The answer (Indefinite) [src]
  /                                                                   
 |                          /     2      \             /  ___        \
 |    x - 1              log\3 + x  + 2*x/     ___     |\/ 2 *(1 + x)|
 | ------------ dx = C + ----------------- - \/ 2 *atan|-------------|
 |  2                            2                     \      2      /
 | x  + 2*x + 3                                                       
 |                                                                    
/                                                                     
$$\int \frac{x - 1}{\left(x^{2} + 2 x\right) + 3}\, dx = C + \frac{\log{\left(x^{2} + 2 x + 3 \right)}}{2} - \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(x + 1\right)}{2} \right)}$$
The graph
The answer [src]
                            /  ___\                    
log(6)   log(3)     ___     |\/ 2 |     ___     /  ___\
------ - ------ + \/ 2 *atan|-----| - \/ 2 *atan\\/ 2 /
  2        2                \  2  /                    
$$- \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)} - \frac{\log{\left(3 \right)}}{2} + \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)} + \frac{\log{\left(6 \right)}}{2}$$
=
=
                            /  ___\                    
log(6)   log(3)     ___     |\/ 2 |     ___     /  ___\
------ - ------ + \/ 2 *atan|-----| - \/ 2 *atan\\/ 2 /
  2        2                \  2  /                    
$$- \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)} - \frac{\log{\left(3 \right)}}{2} + \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)} + \frac{\log{\left(6 \right)}}{2}$$
log(6)/2 - log(3)/2 + sqrt(2)*atan(sqrt(2)/2) - sqrt(2)*atan(sqrt(2))
Numerical answer [src]
-0.134028376065004
-0.134028376065004
The graph
Integral of (x-1)/(x^2+2x+3) dx

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