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x/(9+x^2)
Solving integrals/ x/(9+x^2)

Integral of x/(9+x^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1          
  /          
 |           
 |    x      
 |  ------ dx
 |       2   
 |  9 + x    
 |           
/            
0
$$\int\limits_{0}^{1} \frac{x}{x^{2} + 9}\, dx$$ 
Integral(x/(9 + x^2), (x, 0, 1))
Detail solution
We have the integral:
  /         
 |          
 |   x      
 | ------ dx
 |      2   
 | 9 + x    
 |          
/
Rewrite the integrand
         /    2*x     \             
         |------------|      /0\    
         | 2          |      |-|    
  x      \x  + 0*x + 9/      \9/    
------ = -------------- + ----------
     2         2               2    
9 + x                     /-x \     
                          |---|  + 1
                          \ 3 /
or
  /           
 |            
 |   x        
 | ------ dx  
 |      2    =
 | 9 + x      
 |            
/             
  
  /               
 |                
 |     2*x        
 | ------------ dx
 |  2             
 | x  + 0*x + 9   
 |                
/                 
------------------
        2
In the integral
  /               
 |                
 |     2*x        
 | ------------ dx
 |  2             
 | x  + 0*x + 9   
 |                
/                 
------------------
        2
do replacement
     2
u = x
then
the integral =
  /                     
 |                      
 |   1                  
 | ----- du             
 | 9 + u                
 |                      
/             log(9 + u)
----------- = ----------
     2            2
do backward replacement
  /                             
 |                              
 |     2*x                      
 | ------------ dx              
 |  2                           
 | x  + 0*x + 9                 
 |                      /     2\
/                    log\9 + x /
------------------ = -----------
        2                 2
In the integral
0
do replacement
    -x 
v = ---
     3
then
the integral =
True
do backward replacement
True
Solution is:
       /     2\
    log\9 + x /
C + -----------
         2
The answer (Indefinite) [src] 
  /                           
 |                    /     2\
 |   x             log\9 + x /
 | ------ dx = C + -----------
 |      2               2     
 | 9 + x                      
 |                            
/
$$\int \frac{x}{x^{2} + 9}\, dx = C + \frac{\log{\left(x^{2} + 9 \right)}}{2}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
log(10)   log(9)
------- - ------
   2        2
$$- \frac{\log{\left(9 \right)}}{2} + \frac{\log{\left(10 \right)}}{2}$$ 
=
= 
log(10)   log(9)
------- - ------
   2        2
$$- \frac{\log{\left(9 \right)}}{2} + \frac{\log{\left(10 \right)}}{2}$$ 
log(10)/2 - log(9)/2
Numerical answer [src] 
0.0526802578289131
0.0526802578289131
The graph
Integral of x/(9+x^2) dx

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

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