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1/(x^2+2*x+5)
Solving integrals/ 1/(x^2+2*x+5)

Integral of 1/(x^2+2*x+5) 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 + 5   
 |                 
/                  
0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 2 x\right) + 5}\, dx$$ 
Integral(1/(x^2 + 2*x + 5), (x, 0, 1))
Detail solution
We have the integral:
  /               
 |                
 |      1         
 | ------------ dx
 |  2             
 | x  + 2*x + 5   
 |                
/
Rewrite the integrand
     1                 1         
------------ = ------------------
 2               /         2    \
x  + 2*x + 5     |/  x   1\     |
               4*||- - - -|  + 1|
                 \\  2   2/     /
or
  /                 
 |                  
 |      1           
 | ------------ dx  
 |  2              =
 | x  + 2*x + 5     
 |                  
/                   
  
  /                 
 |                  
 |       1          
 | -------------- dx
 |          2       
 | /  x   1\        
 | |- - - -|  + 1   
 | \  2   2/        
 |                  
/                   
--------------------
         4
In the integral
  /                 
 |                  
 |       1          
 | -------------- dx
 |          2       
 | /  x   1\        
 | |- - - -|  + 1   
 | \  2   2/        
 |                  
/                   
--------------------
         4
do replacement
      1   x
v = - - - -
      2   2
then
the integral =
  /                   
 |                    
 |   1                
 | ------ dv          
 |      2             
 | 1 + v              
 |                    
/              atan(v)
------------ = -------
     4            4
do backward replacement
  /                               
 |                                
 |       1                        
 | -------------- dx              
 |          2                     
 | /  x   1\                      
 | |- - - -|  + 1                 
 | \  2   2/               /1   x\
 |                     atan|- + -|
/                          \2   2/
-------------------- = -----------
         4                  2
Solution is:
        /1   x\
    atan|- + -|
        \2   2/
C + -----------
         2
The answer (Indefinite) [src] 
  /                          /1   x\
 |                       atan|- + -|
 |      1                    \2   2/
 | ------------ dx = C + -----------
 |  2                         2     
 | x  + 2*x + 5                     
 |                                  
/
$$\int \frac{1}{\left(x^{2} + 2 x\right) + 5}\, dx = C + \frac{\operatorname{atan}{\left(\frac{x}{2} + \frac{1}{2} \right)}}{2}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
  atan(1/2)   pi
- --------- + --
      2       8
$$- \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2} + \frac{\pi}{8}$$ 
=
= 
  atan(1/2)   pi
- --------- + --
      2       8
$$- \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2} + \frac{\pi}{8}$$ 
-atan(1/2)/2 + pi/8
Numerical answer [src] 
0.160875277198321
0.160875277198321
The graph
Integral of 1/(x^2+2*x+5) dx

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

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