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1/(2+x^2)

Integral of 1/(2+x^2) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

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  1          
  /          
 |           
 |    1      
 |  ------ dx
 |       2   
 |  2 + x    
 |           
/            
0            
$$\int\limits_{0}^{1} \frac{1}{x^{2} + 2}\, dx$$
Integral(1/(2 + x^2), (x, 0, 1))
Detail solution
We have the integral:
  /         
 |          
 |   1      
 | ------ dx
 |      2   
 | 2 + x    
 |          
/           
Rewrite the integrand
  1               1          
------ = --------------------
     2     /           2    \
2 + x      |/   ___   \     |
           ||-\/ 2    |     |
         2*||-------*x|  + 1|
           \\   2     /     /
or
  /           
 |            
 |   1        
 | ------ dx  
 |      2    =
 | 2 + x      
 |            
/             
  
  /                   
 |                    
 |        1           
 | ---------------- dx
 |            2       
 | /   ___   \        
 | |-\/ 2    |        
 | |-------*x|  + 1   
 | \   2     /        
 |                    
/                     
----------------------
          2           
In the integral
  /                   
 |                    
 |        1           
 | ---------------- dx
 |            2       
 | /   ___   \        
 | |-\/ 2    |        
 | |-------*x|  + 1   
 | \   2     /        
 |                    
/                     
----------------------
          2           
do replacement
         ___ 
    -x*\/ 2  
v = ---------
        2    
then
the integral =
  /                   
 |                    
 |   1                
 | ------ dv          
 |      2             
 | 1 + v              
 |                    
/              atan(v)
------------ = -------
     2            2   
do backward replacement
  /                                         
 |                                          
 |        1                                 
 | ---------------- dx                      
 |            2                             
 | /   ___   \                              
 | |-\/ 2    |                              
 | |-------*x|  + 1                /    ___\
 | \   2     /             ___     |x*\/ 2 |
 |                       \/ 2 *atan|-------|
/                                  \   2   /
---------------------- = -------------------
          2                       2         
Solution is:
              /    ___\
      ___     |x*\/ 2 |
    \/ 2 *atan|-------|
              \   2   /
C + -------------------
             2         
The answer (Indefinite) [src]
                             /    ___\
  /                  ___     |x*\/ 2 |
 |                 \/ 2 *atan|-------|
 |   1                       \   2   /
 | ------ dx = C + -------------------
 |      2                   2         
 | 2 + x                              
 |                                    
/                                     
$$\int \frac{1}{x^{2} + 2}\, dx = C + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}$$
The graph
The answer [src]
          /  ___\
  ___     |\/ 2 |
\/ 2 *atan|-----|
          \  2  /
-----------------
        2        
$$\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2}$$
=
=
          /  ___\
  ___     |\/ 2 |
\/ 2 *atan|-----|
          \  2  /
-----------------
        2        
$$\frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2}$$
sqrt(2)*atan(sqrt(2)/2)/2
Numerical answer [src]
0.435209875683552
0.435209875683552
The graph
Integral of 1/(2+x^2) dx

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