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1/(x^2+2x+3)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |   2             
 |  x  + 2*x + 3   
 |                 
/                  
0                  
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 2 x\right) + 3}\, dx$$
Integral(1/(x^2 + 2*x + 3), (x, 0, 1))
Detail solution
We have the integral:
  /               
 |                
 |      1         
 | ------------ dx
 |  2             
 | x  + 2*x + 3   
 |                
/                 
Rewrite the integrand
     1                      1              
------------ = ----------------------------
 2               /                   2    \
x  + 2*x + 3     |/   ___        ___\     |
                 ||-\/ 2       \/ 2 |     |
               2*||-------*x - -----|  + 1|
                 \\   2          2  /     /
or
  /                 
 |                  
 |      1           
 | ------------ dx  
 |  2              =
 | x  + 2*x + 3     
 |                  
/                   
  
  /                           
 |                            
 |            1               
 | ------------------------ dx
 |                    2       
 | /   ___        ___\        
 | |-\/ 2       \/ 2 |        
 | |-------*x - -----|  + 1   
 | \   2          2  /        
 |                            
/                             
------------------------------
              2               
In the integral
  /                           
 |                            
 |            1               
 | ------------------------ dx
 |                    2       
 | /   ___        ___\        
 | |-\/ 2       \/ 2 |        
 | |-------*x - -----|  + 1   
 | \   2          2  /        
 |                            
/                             
------------------------------
              2               
do replacement
        ___       ___
      \/ 2    x*\/ 2 
v = - ----- - -------
        2        2   
then
the integral =
  /                   
 |                    
 |   1                
 | ------ dv          
 |      2             
 | 1 + v              
 |                    
/              atan(v)
------------ = -------
     2            2   
do backward replacement
  /                                                         
 |                                                          
 |            1                                             
 | ------------------------ dx                              
 |                    2                                     
 | /   ___        ___\                                      
 | |-\/ 2       \/ 2 |                                      
 | |-------*x - -----|  + 1                /  ___       ___\
 | \   2          2  /             ___     |\/ 2    x*\/ 2 |
 |                               \/ 2 *atan|----- + -------|
/                                          \  2        2   /
------------------------------ = ---------------------------
              2                               2             
Solution is:
              /  ___       ___\
      ___     |\/ 2    x*\/ 2 |
    \/ 2 *atan|----- + -------|
              \  2        2   /
C + ---------------------------
                 2             
The answer (Indefinite) [src]
                                   /  ___        \
  /                        ___     |\/ 2 *(1 + x)|
 |                       \/ 2 *atan|-------------|
 |      1                          \      2      /
 | ------------ dx = C + -------------------------
 |  2                                2            
 | x  + 2*x + 3                                   
 |                                                
/                                                 
$$\int \frac{1}{\left(x^{2} + 2 x\right) + 3}\, dx = C + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(x + 1\right)}{2} \right)}}{2}$$
The graph
The answer [src]
                              /  ___\
                      ___     |\/ 2 |
  ___     /  ___\   \/ 2 *atan|-----|
\/ 2 *atan\\/ 2 /             \  2  /
----------------- - -----------------
        2                   2        
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}}{2}$$
=
=
                              /  ___\
                      ___     |\/ 2 |
  ___     /  ___\   \/ 2 *atan|-----|
\/ 2 *atan\\/ 2 /             \  2  /
----------------- - -----------------
        2                   2        
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}}{2}$$
sqrt(2)*atan(sqrt(2))/2 - sqrt(2)*atan(sqrt(2)/2)/2
Numerical answer [src]
0.240300983172488
0.240300983172488
The graph
Integral of 1/(x^2+2x+3) dx

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