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1/(x^2+4x+8)
Solving integrals/ 1/(x^2+4x+8)

Integral of 1/(x^2+4x+8) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |   2             
 |  x  + 4*x + 8   
 |                 
/                  
0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 4 x\right) + 8}\, dx$$ 
Integral(1/(x^2 + 4*x + 8), (x, 0, 1))
Detail solution
We have the integral:
  /               
 |                
 |      1         
 | ------------ dx
 |  2             
 | x  + 4*x + 8   
 |                
/
Rewrite the integrand
     1                 1         
------------ = ------------------
 2               /         2    \
x  + 4*x + 8     |/  x    \     |
               4*||- - - 1|  + 1|
                 \\  2    /     /
or
  /                 
 |                  
 |      1           
 | ------------ dx  
 |  2              =
 | x  + 4*x + 8     
 |                  
/                   
  
  /                 
 |                  
 |       1          
 | -------------- dx
 |          2       
 | /  x    \        
 | |- - - 1|  + 1   
 | \  2    /        
 |                  
/                   
--------------------
         4
In the integral
  /                 
 |                  
 |       1          
 | -------------- dx
 |          2       
 | /  x    \        
 | |- - - 1|  + 1   
 | \  2    /        
 |                  
/                   
--------------------
         4
do replacement
         x
v = -1 - -
         2
then
the integral =
  /                   
 |                    
 |   1                
 | ------ dv          
 |      2             
 | 1 + v              
 |                    
/              atan(v)
------------ = -------
     4            4
do backward replacement
  /                               
 |                                
 |       1                        
 | -------------- dx              
 |          2                     
 | /  x    \                      
 | |- - - 1|  + 1                 
 | \  2    /               /    x\
 |                     atan|1 + -|
/                          \    2/
-------------------- = -----------
         4                  2
Solution is:
        /    x\
    atan|1 + -|
        \    2/
C + -----------
         2
The answer (Indefinite) [src] 
  /                          /    x\
 |                       atan|1 + -|
 |      1                    \    2/
 | ------------ dx = C + -----------
 |  2                         2     
 | x  + 4*x + 8                     
 |                                  
/
$$\int \frac{1}{\left(x^{2} + 4 x\right) + 8}\, dx = C + \frac{\operatorname{atan}{\left(\frac{x}{2} + 1 \right)}}{2}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
atan(3/2)   pi
--------- - --
    2       8
$$- \frac{\pi}{8} + \frac{\operatorname{atan}{\left(\frac{3}{2} \right)}}{2}$$ 
=
= 
atan(3/2)   pi
--------- - --
    2       8
$$- \frac{\pi}{8} + \frac{\operatorname{atan}{\left(\frac{3}{2} \right)}}{2}$$ 
atan(3/2)/2 - pi/8
Numerical answer [src] 
0.0986977799249404
0.0986977799249404
The graph
Integral of 1/(x^2+4x+8) dx

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

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