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Integral of dx/(x²-2x+4) 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 + 4   
 |                 
/                  
0                  
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} - 2 x\right) + 4}\, dx$$
Integral(1/(x^2 - 2*x + 4), (x, 0, 1))
Detail solution
We have the integral:
  /               
 |                
 |      1         
 | ------------ dx
 |  2             
 | x  - 2*x + 4   
 |                
/                 
Rewrite the integrand
     1                      1              
------------ = ----------------------------
 2               /                   2    \
x  - 2*x + 4     |/   ___        ___\     |
                 ||-\/ 3       \/ 3 |     |
               3*||-------*x + -----|  + 1|
                 \\   3          3  /     /
or
  /                 
 |                  
 |      1           
 | ------------ dx  
 |  2              =
 | x  - 2*x + 4     
 |                  
/                   
  
  /                           
 |                            
 |            1               
 | ------------------------ dx
 |                    2       
 | /   ___        ___\        
 | |-\/ 3       \/ 3 |        
 | |-------*x + -----|  + 1   
 | \   3          3  /        
 |                            
/                             
------------------------------
              3               
In the integral
  /                           
 |                            
 |            1               
 | ------------------------ dx
 |                    2       
 | /   ___        ___\        
 | |-\/ 3       \/ 3 |        
 | |-------*x + -----|  + 1   
 | \   3          3  /        
 |                            
/                             
------------------------------
              3               
do replacement
      ___       ___
    \/ 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       \/ 3 |                                        
 | |-------*x + -----|  + 1                /    ___       ___\
 | \   3          3  /             ___     |  \/ 3    x*\/ 3 |
 |                               \/ 3 *atan|- ----- + -------|
/                                          \    3        3   /
------------------------------ = -----------------------------
              3                                3              
Solution is:
              /    ___       ___\
      ___     |  \/ 3    x*\/ 3 |
    \/ 3 *atan|- ----- + -------|
              \    3        3   /
C + -----------------------------
                  3              
The answer (Indefinite) [src]
                                   /  ___         \
  /                        ___     |\/ 3 *(-1 + x)|
 |                       \/ 3 *atan|--------------|
 |      1                          \      3       /
 | ------------ dx = C + --------------------------
 |  2                                3             
 | x  - 2*x + 4                                    
 |                                                 
/                                                  
$$\int \frac{1}{\left(x^{2} - 2 x\right) + 4}\, dx = C + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x - 1\right)}{3} \right)}}{3}$$
The graph
The answer [src]
     ___
pi*\/ 3 
--------
   18   
$$\frac{\sqrt{3} \pi}{18}$$
=
=
     ___
pi*\/ 3 
--------
   18   
$$\frac{\sqrt{3} \pi}{18}$$
pi*sqrt(3)/18
Numerical answer [src]
0.302299894039036
0.302299894039036

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