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Integral of dx/(x^2-4x+5) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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  2                
  /                
 |                 
 |       1         
 |  ------------ dx
 |   2             
 |  x  - 4*x + 5   
 |                 
/                  
0                  
$$\int\limits_{0}^{2} \frac{1}{\left(x^{2} - 4 x\right) + 5}\, dx$$
Integral(1/(x^2 - 4*x + 5), (x, 0, 2))
Detail solution
We have the integral:
  /               
 |                
 |      1         
 | ------------ dx
 |  2             
 | x  - 4*x + 5   
 |                
/                 
Rewrite the integrand
     1                 1        
------------ = -----------------
 2               /        2    \
x  - 4*x + 5   1*\(-x + 2)  + 1/
or
  /                 
 |                  
 |      1           
 | ------------ dx  
 |  2              =
 | x  - 4*x + 5     
 |                  
/                   
  
  /                
 |                 
 |       1         
 | ------------- dx
 |         2       
 | (-x + 2)  + 1   
 |                 
/                  
In the integral
  /                
 |                 
 |       1         
 | ------------- dx
 |         2       
 | (-x + 2)  + 1   
 |                 
/                  
do replacement
v = 2 - x
then
the integral =
  /                   
 |                    
 |   1                
 | ------ dv = atan(v)
 |      2             
 | 1 + v              
 |                    
/                     
do backward replacement
  /                               
 |                                
 |       1                        
 | ------------- dx = atan(-2 + x)
 |         2                      
 | (-x + 2)  + 1                  
 |                                
/                                 
Solution is:
C + atan(-2 + x)
The answer (Indefinite) [src]
  /                                  
 |                                   
 |      1                            
 | ------------ dx = C + atan(-2 + x)
 |  2                                
 | x  - 4*x + 5                      
 |                                   
/                                    
$$\int \frac{1}{\left(x^{2} - 4 x\right) + 5}\, dx = C + \operatorname{atan}{\left(x - 2 \right)}$$
The graph
The answer [src]
atan(2)
$$\operatorname{atan}{\left(2 \right)}$$
=
=
atan(2)
$$\operatorname{atan}{\left(2 \right)}$$
atan(2)
Numerical answer [src]
1.10714871779409
1.10714871779409

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