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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1          
  /          
 |           
 |    x      
 |  ------ dx
 |   2       
 |  x  + x   
 |           
/            
0
$$\int\limits_{0}^{1} \frac{x}{x^{2} + x}\, dx$$ 
Integral(x/(x^2 + x), (x, 0, 1))
Detail solution
We have the integral:
  /         
 |          
 |   x      
 | ------ dx
 |  2       
 | x  + x   
 |          
/
Rewrite the integrand
         /2*x + 1\
         |-------|
         |  2    |
  x      \ x  + x/
------ = ---------
 2           2    
x  + x
or
  /           
 |            
 |   x        
 | ------ dx  
 |  2        =
 | x  + x     
 |            
/             
  
  /          
 |           
 | 2*x + 1   
 | ------- dx
 |   2       
 |  x  + x   
 |           
/            
-------------
      2
In the integral
  /          
 |           
 | 2*x + 1   
 | ------- dx
 |   2       
 |  x  + x   
 |           
/            
-------------
      2
do replacement
         2
u = x + x
then
the integral =
  /             
 |              
 | 1            
 | - du         
 | u            
 |              
/         log(u)
------- = ------
   2        2
do backward replacement
  /                        
 |                         
 | 2*x + 1                 
 | ------- dx              
 |   2                     
 |  x  + x                 
 |                 /     2\
/               log\x + x /
------------- = -----------
      2              2
Solution is:
C + log(1 + x)
The answer (Indefinite) [src] 
  /                                                     
 |                                   /     2\           
 |   x             log(2 + 2*x)   log\x + x /   log(2*x)
 | ------ dx = C + ------------ + ----------- - --------
 |  2                   2              2           2    
 | x  + x                                               
 |                                                      
/
$$\int \frac{x}{x^{2} + x}\, dx = C - \frac{\log{\left(2 x \right)}}{2} + \frac{\log{\left(2 x + 2 \right)}}{2} + \frac{\log{\left(x^{2} + x \right)}}{2}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
log(2)
$$\log{\left(2 \right)}$$ 
=
= 
log(2)
$$\log{\left(2 \right)}$$ 
log(2)
Numerical answer [src] 
0.693147180559945
0.693147180559945
The graph
Integral of x/(x^2+x) dx

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

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