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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 oo          
  /          
 |           
 |  x - 2    
 |  ------ dx
 |   2       
 |  x  + 1   
 |           
/            
-oo
$$\int\limits_{-\infty}^{\infty} \frac{x - 2}{x^{2} + 1}\, dx$$ 
Integral((x - 2)/(x^2 + 1), (x, -oo, oo))
Detail solution
We have the integral:
  /         
 |          
 | x - 2    
 | ------ dx
 |  2       
 | x  + 1   
 |          
/
Rewrite the integrand
         /    2*x     \            
         |------------|     /-2 \  
         | 2          |     |---|  
x - 2    \x  + 0*x + 1/     \ 1 /  
------ = -------------- + ---------
 2             2              2    
x  + 1                    (-x)  + 1
or
  /           
 |            
 | x - 2      
 | ------ dx  
 |  2        =
 | x  + 1     
 |            
/             
  
  /                                   
 |                                    
 |     2*x                            
 | ------------ dx                    
 |  2                                 
 | x  + 0*x + 1          /            
 |                      |             
/                       |     1       
------------------ - 2* | --------- dx
        2               |     2       
                        | (-x)  + 1   
                        |             
                       /
In the integral
  /               
 |                
 |     2*x        
 | ------------ dx
 |  2             
 | x  + 0*x + 1   
 |                
/                 
------------------
        2
do replacement
     2
u = x
then
the integral =
  /                     
 |                      
 |   1                  
 | ----- du             
 | 1 + u                
 |                      
/             log(1 + u)
----------- = ----------
     2            2
do backward replacement
  /                             
 |                              
 |     2*x                      
 | ------------ dx              
 |  2                           
 | x  + 0*x + 1                 
 |                      /     2\
/                    log\1 + x /
------------------ = -----------
        2                 2
In the integral
     /            
    |             
    |     1       
-2* | --------- dx
    |     2       
    | (-x)  + 1   
    |             
   /
do replacement
v = -x
then
the integral =
     /                      
    |                       
    |   1                   
-2* | ------ dv = -2*atan(v)
    |      2                
    | 1 + v                 
    |                       
   /
do backward replacement
     /                         
    |                          
    |     1                    
-2* | --------- dx = -2*atan(x)
    |     2                    
    | (-x)  + 1                
    |                          
   /
Solution is:
       /     2\            
    log\1 + x /            
C + ----------- - 2*atan(x)
         2
The answer (Indefinite) [src] 
  /                                       
 |                    /     2\            
 | x - 2           log\1 + x /            
 | ------ dx = C + ----------- - 2*atan(x)
 |  2                   2                 
 | x  + 1                                 
 |                                        
/
$$\int \frac{x - 2}{x^{2} + 1}\, dx = C + \frac{\log{\left(x^{2} + 1 \right)}}{2} - 2 \operatorname{atan}{\left(x \right)}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
nan
$$\text{NaN}$$ 
=
= 
nan
$$\text{NaN}$$ 
nan
Numerical answer [src] 
-6.28318530717959
-6.28318530717959

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

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