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Integral of x/(x^2+y^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

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