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x/(x^2+4x+5)

Integral of x/(x^2+4x+5) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                
  /                
 |                 
 |       x         
 |  ------------ dx
 |   2             
 |  x  + 4*x + 5   
 |                 
/                  
0                  
$$\int\limits_{0}^{1} \frac{x}{x^{2} + 4 x + 5}\, dx$$
Integral(x/(x^2 + 4*x + 5), (x, 0, 1))
Detail solution
We have the integral:
  /                 
 |                  
 |        x         
 | 1*------------ dx
 |    2             
 |   x  + 4*x + 5   
 |                  
/                   
Rewrite the integrand
               /  1*2*x + 4   \                
               |--------------|       /-2 \    
               |   2          |       |---|    
     x         \1*x  + 4*x + 5/       \ 1 /    
------------ = ---------------- + -------------
 2                    2                   2    
x  + 4*x + 5                      (-x - 2)  + 1
or
  /                   
 |                    
 |        x           
 | 1*------------ dx  
 |    2              =
 |   x  + 4*x + 5     
 |                    
/                     
  
  /                                         
 |                                          
 |   1*2*x + 4                              
 | -------------- dx                        
 |    2                                     
 | 1*x  + 4*x + 5          /                
 |                        |                 
/                         |       1         
-------------------- - 2* | ------------- dx
         2                |         2       
                          | (-x - 2)  + 1   
                          |                 
                         /                  
In the integral
  /                 
 |                  
 |   1*2*x + 4      
 | -------------- dx
 |    2             
 | 1*x  + 4*x + 5   
 |                  
/                   
--------------------
         2          
do replacement
     2      
u = x  + 4*x
then
the integral =
  /                     
 |                      
 |   1                  
 | ----- du             
 | 5 + u                
 |                      
/             log(5 + u)
----------- = ----------
     2            2     
do backward replacement
  /                                     
 |                                      
 |   1*2*x + 4                          
 | -------------- dx                    
 |    2                                 
 | 1*x  + 4*x + 5                       
 |                        /     2      \
/                      log\5 + x  + 4*x/
-------------------- = -----------------
         2                     2        
In the integral
     /                
    |                 
    |       1         
-2* | ------------- dx
    |         2       
    | (-x - 2)  + 1   
    |                 
   /                  
do replacement
v = -2 - x
then
the integral =
     /                      
    |                       
    |   1                   
-2* | ------ dv = -2*atan(v)
    |      2                
    | 1 + v                 
    |                       
   /                        
do backward replacement
     /                                 
    |                                  
    |       1                          
-2* | ------------- dx = -2*atan(2 + x)
    |         2                        
    | (-x - 2)  + 1                    
    |                                  
   /                                   
Solution is:
       /     2      \                
    log\5 + x  + 4*x/                
C + ----------------- - 2*atan(2 + x)
            2                        
The answer (Indefinite) [src]
  /                                                       
 |                          /     2      \                
 |      x                log\5 + x  + 4*x/                
 | ------------ dx = C + ----------------- - 2*atan(2 + x)
 |  2                            2                        
 | x  + 4*x + 5                                           
 |                                                        
/                                                         
$${{\log \left(x^2+4\,x+5\right)}\over{2}}-2\,\arctan \left({{2\,x+4 }\over{2}}\right)$$
The graph
The answer [src]
log(10)                           log(5)
------- - 2*atan(3) + 2*atan(2) - ------
   2                                2   
$${{\log 10}\over{2}}-{{\log 5}\over{2}}-2\,\arctan 3+2\,\arctan 2$$
=
=
log(10)                           log(5)
------- - 2*atan(3) + 2*atan(2) - ------
   2                                2   
$$- 2 \operatorname{atan}{\left(3 \right)} - \frac{\log{\left(5 \right)}}{2} + \frac{\log{\left(10 \right)}}{2} + 2 \operatorname{atan}{\left(2 \right)}$$
Numerical answer [src]
0.0627794810716448
0.0627794810716448
The graph
Integral of x/(x^2+4x+5) dx

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