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Integral of 1/(x^2+4x+7) dx

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

You have entered [src]
  1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |   2             
 |  x  + 4*x + 7   
 |                 
/                  
-3                 
$$\int\limits_{-3}^{1} \frac{1}{\left(x^{2} + 4 x\right) + 7}\, dx$$
Integral(1/(x^2 + 4*x + 7), (x, -3, 1))
Detail solution
We have the integral:
  /               
 |                
 |      1         
 | ------------ dx
 |  2             
 | x  + 4*x + 7   
 |                
/                 
Rewrite the integrand
     1                       1               
------------ = ------------------------------
 2               /                     2    \
x  + 4*x + 7     |/   ___          ___\     |
                 ||-\/ 3       2*\/ 3 |     |
               3*||-------*x - -------|  + 1|
                 \\   3           3   /     /
or
  /                 
 |                  
 |      1           
 | ------------ dx  
 |  2              =
 | x  + 4*x + 7     
 |                  
/                   
  
  /                             
 |                              
 |             1                
 | -------------------------- dx
 |                      2       
 | /   ___          ___\        
 | |-\/ 3       2*\/ 3 |        
 | |-------*x - -------|  + 1   
 | \   3           3   /        
 |                              
/                               
--------------------------------
               3                
In the integral
  /                             
 |                              
 |             1                
 | -------------------------- dx
 |                      2       
 | /   ___          ___\        
 | |-\/ 3       2*\/ 3 |        
 | |-------*x - -------|  + 1   
 | \   3           3   /        
 |                              
/                               
--------------------------------
               3                
do replacement
          ___       ___
      2*\/ 3    x*\/ 3 
v = - ------- - -------
         3         3   
then
the integral =
  /                   
 |                    
 |   1                
 | ------ dv          
 |      2             
 | 1 + v              
 |                    
/              atan(v)
------------ = -------
     3            3   
do backward replacement
  /                                                             
 |                                                              
 |             1                                                
 | -------------------------- dx                                
 |                      2                                       
 | /   ___          ___\                                        
 | |-\/ 3       2*\/ 3 |                                        
 | |-------*x - -------|  + 1                /    ___       ___\
 | \   3           3   /             ___     |2*\/ 3    x*\/ 3 |
 |                                 \/ 3 *atan|------- + -------|
/                                            \   3         3   /
-------------------------------- = -----------------------------
               3                                 3              
Solution is:
              /    ___       ___\
      ___     |2*\/ 3    x*\/ 3 |
    \/ 3 *atan|------- + -------|
              \   3         3   /
C + -----------------------------
                  3              
The answer (Indefinite) [src]
                                   /  ___        \
  /                        ___     |\/ 3 *(2 + x)|
 |                       \/ 3 *atan|-------------|
 |      1                          \      3      /
 | ------------ dx = C + -------------------------
 |  2                                3            
 | x  + 4*x + 7                                   
 |                                                
/                                                 
$$\int \frac{1}{\left(x^{2} + 4 x\right) + 7}\, dx = C + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x + 2\right)}{3} \right)}}{3}$$
The graph
The answer [src]
     ___
pi*\/ 3 
--------
   6    
$$\frac{\sqrt{3} \pi}{6}$$
=
=
     ___
pi*\/ 3 
--------
   6    
$$\frac{\sqrt{3} \pi}{6}$$
pi*sqrt(3)/6
Numerical answer [src]
0.906899682117109
0.906899682117109

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