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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]
  1          
  /          
 |           
 |  x + 2    
 |  ------ dx
 |   2       
 |  x  + 1   
 |           
/            
0            
$$\int\limits_{0}^{1} \frac{x + 2}{x^{2} + 1}\, dx$$
Integral((x + 2)/(x^2 + 1), (x, 0, 1))
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                                 
 |                                        
/                                         
$$\int \frac{x + 2}{x^{2} + 1}\, dx = C + \frac{\log{\left(x^{2} + 1 \right)}}{2} + 2 \operatorname{atan}{\left(x \right)}$$
The graph
The answer [src]
pi   log(2)
-- + ------
2      2   
$$\frac{\log{\left(2 \right)}}{2} + \frac{\pi}{2}$$
=
=
pi   log(2)
-- + ------
2      2   
$$\frac{\log{\left(2 \right)}}{2} + \frac{\pi}{2}$$
pi/2 + log(2)/2
Numerical answer [src]
1.91736991707487
1.91736991707487

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