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1/(x^2+2x+10)
Solving integrals/ 1/(x^2+2x+10)

Integral of 1/(x^2+2x+10) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
 oo                 
  /                 
 |                  
 |        1         
 |  ------------- dx
 |   2              
 |  x  + 2*x + 10   
 |                  
/                   
1
$$\int\limits_{1}^{\infty} \frac{1}{\left(x^{2} + 2 x\right) + 10}\, dx$$ 
Integral(1/(x^2 + 2*x + 10), (x, 1, oo))
Detail solution
We have the integral:
  /                
 |                 
 |       1         
 | ------------- dx
 |  2              
 | x  + 2*x + 10   
 |                 
/
Rewrite the integrand
      1                 1         
------------- = ------------------
 2                /         2    \
x  + 2*x + 10     |/  x   1\     |
                9*||- - - -|  + 1|
                  \\  3   3/     /
or
  /                  
 |                   
 |       1           
 | ------------- dx  
 |  2               =
 | x  + 2*x + 10     
 |                   
/                    
  
  /                 
 |                  
 |       1          
 | -------------- dx
 |          2       
 | /  x   1\        
 | |- - - -|  + 1   
 | \  3   3/        
 |                  
/                   
--------------------
         9
In the integral
  /                 
 |                  
 |       1          
 | -------------- dx
 |          2       
 | /  x   1\        
 | |- - - -|  + 1   
 | \  3   3/        
 |                  
/                   
--------------------
         9
do replacement
      1   x
v = - - - -
      3   3
then
the integral =
  /                   
 |                    
 |   1                
 | ------ dv          
 |      2             
 | 1 + v              
 |                    
/              atan(v)
------------ = -------
     9            9
do backward replacement
  /                               
 |                                
 |       1                        
 | -------------- dx              
 |          2                     
 | /  x   1\                      
 | |- - - -|  + 1                 
 | \  3   3/               /1   x\
 |                     atan|- + -|
/                          \3   3/
-------------------- = -----------
         9                  3
Solution is:
        /1   x\
    atan|- + -|
        \3   3/
C + -----------
         3
The answer (Indefinite) [src] 
  /                           /1   x\
 |                        atan|- + -|
 |       1                    \3   3/
 | ------------- dx = C + -----------
 |  2                          3     
 | x  + 2*x + 10                     
 |                                   
/
$$\int \frac{1}{\left(x^{2} + 2 x\right) + 10}\, dx = C + \frac{\operatorname{atan}{\left(\frac{x}{3} + \frac{1}{3} \right)}}{3}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
  atan(2/3)   pi
- --------- + --
      3       6
$$- \frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{3} + \frac{\pi}{6}$$ 
=
= 
  atan(2/3)   pi
- --------- + --
      3       6
$$- \frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{3} + \frac{\pi}{6}$$ 
-atan(2/3)/3 + pi/6
The graph
Integral of 1/(x^2+2x+10) dx

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

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