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1/(x^2+3)

Integral of 1/(x^2+3) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1          
  /          
 |           
 |    1      
 |  ------ dx
 |   2       
 |  x  + 3   
 |           
/            
0            
$$\int\limits_{0}^{1} \frac{1}{x^{2} + 3}\, dx$$
Integral(1/(x^2 + 3), (x, 0, 1))
Detail solution
We have the integral:
  /         
 |          
 |   1      
 | ------ dx
 |  2       
 | x  + 3   
 |          
/           
Rewrite the integrand
  1               1          
------ = --------------------
 2         /           2    \
x  + 3     |/   ___   \     |
           ||-\/ 3    |     |
         3*||-------*x|  + 1|
           \\   3     /     /
or
  /           
 |            
 |   1        
 | ------ dx  
 |  2        =
 | x  + 3     
 |            
/             
  
  /                   
 |                    
 |        1           
 | ---------------- dx
 |            2       
 | /   ___   \        
 | |-\/ 3    |        
 | |-------*x|  + 1   
 | \   3     /        
 |                    
/                     
----------------------
          3           
In the integral
  /                   
 |                    
 |        1           
 | ---------------- dx
 |            2       
 | /   ___   \        
 | |-\/ 3    |        
 | |-------*x|  + 1   
 | \   3     /        
 |                    
/                     
----------------------
          3           
do replacement
         ___ 
    -x*\/ 3  
v = ---------
        3    
then
the integral =
  /                   
 |                    
 |   1                
 | ------ dv          
 |      2             
 | 1 + v              
 |                    
/              atan(v)
------------ = -------
     3            3   
do backward replacement
  /                                         
 |                                          
 |        1                                 
 | ---------------- dx                      
 |            2                             
 | /   ___   \                              
 | |-\/ 3    |                              
 | |-------*x|  + 1                /    ___\
 | \   3     /             ___     |x*\/ 3 |
 |                       \/ 3 *atan|-------|
/                                  \   3   /
---------------------- = -------------------
          3                       3         
Solution is:
              /    ___\
      ___     |x*\/ 3 |
    \/ 3 *atan|-------|
              \   3   /
C + -------------------
             3         
The answer (Indefinite) [src]
                             /    ___\
  /                  ___     |x*\/ 3 |
 |                 \/ 3 *atan|-------|
 |   1                       \   3   /
 | ------ dx = C + -------------------
 |  2                       3         
 | x  + 3                             
 |                                    
/                                     
$$\int \frac{1}{x^{2} + 3}\, dx = C + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}}{3}$$
The graph
The answer [src]
     ___
pi*\/ 3 
--------
   18   
$$\frac{\sqrt{3} \pi}{18}$$
=
=
     ___
pi*\/ 3 
--------
   18   
$$\frac{\sqrt{3} \pi}{18}$$
pi*sqrt(3)/18
Numerical answer [src]
0.302299894039036
0.302299894039036
The graph
Integral of 1/(x^2+3) dx

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