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Solving integrals/ 3*x/(2+x^2)

Integral of 3*x/(2+x^2) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1          
  /          
 |           
 |   3*x     
 |  ------ dx
 |       2   
 |  2 + x    
 |           
/            
-1
$$\int\limits_{-1}^{1} \frac{3 x}{x^{2} + 2}\, dx$$ 
Integral((3*x)/(2 + x^2), (x, -1, 1))
Detail solution
We have the integral:
  /         
 |          
 |  3*x     
 | ------ dx
 |      2   
 | 2 + x    
 |          
/
Rewrite the integrand
               2*x                        
         3*------------         /0\       
            2                   |-|       
 3*x       x  + 0*x + 2         \2/       
------ = -------------- + ----------------
     2         2                     2    
2 + x                     /   ___   \     
                          |-\/ 2    |     
                          |-------*x|  + 1
                          \   2     /
or
  /           
 |            
 |  3*x       
 | ------ dx  
 |      2    =
 | 2 + x      
 |            
/             
  
    /               
   |                
   |     2*x        
3* | ------------ dx
   |  2             
   | x  + 0*x + 2   
   |                
  /                 
--------------------
         2
In the integral
    /               
   |                
   |     2*x        
3* | ------------ dx
   |  2             
   | x  + 0*x + 2   
   |                
  /                 
--------------------
         2
do replacement
     2
u = x
then
the integral =
    /                       
   |                        
   |   1                    
3* | ----- du               
   | 2 + u                  
   |                        
  /             3*log(2 + u)
------------- = ------------
      2              2
do backward replacement
    /                               
   |                                
   |     2*x                        
3* | ------------ dx                
   |  2                             
   | x  + 0*x + 2                   
   |                        /     2\
  /                    3*log\2 + x /
-------------------- = -------------
         2                   2
In the integral
0
do replacement
         ___ 
    -x*\/ 2  
v = ---------
        2
then
the integral =
True
do backward replacement
True
Solution is:
         /     2\
    3*log\2 + x /
C + -------------
          2
The answer (Indefinite) [src] 
  /                             
 |                      /     2\
 |  3*x            3*log\2 + x /
 | ------ dx = C + -------------
 |      2                2      
 | 2 + x                        
 |                              
/
$$\int \frac{3 x}{x^{2} + 2}\, dx = C + \frac{3 \log{\left(x^{2} + 2 \right)}}{2}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
0
$$0$$ 
=
= 
0
$$0$$ 
0
Numerical answer [src] 
0.0
0.0

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

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