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Solving integrals/ (ax+b)÷(ax^2+2bx+c)

Integral of (ax+b)÷(ax^2+2bx+c) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1                    
  /                    
 |                     
 |      a*x + b        
 |  ---------------- dx
 |     2               
 |  a*x  + 2*b*x + c   
 |                     
/                      
0
$$\int\limits_{0}^{1} \frac{a x + b}{a x^{2} + 2 b x + c}\, dx$$
Simplify
Detail solution
We have the integral:
  /                     
 |                      
 |       a*x + b        
 | 1*---------------- dx
 |      2               
 |   a*x  + 2*b*x + c   
 |                      
/
Rewrite the integrand
                                                           /     0      \                  
                                                           |------------|                  
                   /  2*a*x + 2*b   \                      |/   2      \|                  
                   |----------------|                      ||- b  + a*c||                  
                   |   2            |                      ||----------||                  
    a*x + b        \a*x  + 2*b*x + c/                      \\    a     //                  
---------------- = ------------------ + ---------------------------------------------------
   2                       2                                                          2    
a*x  + 2*b*x + c                        /        ____________            ____________\     
                                        |       /     1                 /     1      |     
                                        |-a*   /  ---------- *x - b*   /  ---------- |  + 1
                                        |     /      2                /      2       |     
                                        \   \/    - b  + a*c        \/    - b  + a*c /
or
  /                       
 |                        
 |       a*x + b          
 | 1*---------------- dx  
 |      2                =
 |   a*x  + 2*b*x + c     
 |                        
/                         
  
  /                   
 |                    
 |   2*a*x + 2*b      
 | ---------------- dx
 |    2               
 | a*x  + 2*b*x + c   
 |                    
/                     
----------------------
          2
In the integral
  /                   
 |                    
 |   2*a*x + 2*b      
 | ---------------- dx
 |    2               
 | a*x  + 2*b*x + c   
 |                    
/                     
----------------------
          2
do replacement
       2        
u = a*x  + 2*b*x
then
the integral =
  /                     
 |                      
 |   1                  
 | ----- du             
 | c + u                
 |                      
/             log(c + u)
----------- = ----------
     2            2
do backward replacement
  /                                           
 |                                            
 |   2*a*x + 2*b                              
 | ---------------- dx                        
 |    2                                       
 | a*x  + 2*b*x + c                           
 |                          /       2        \
/                        log\c + a*x  + 2*b*x/
---------------------- = ---------------------
          2                        2
In the integral
0
do replacement
             ____________            ____________
            /     1                 /     1      
v = - b*   /  ----------  - a*x*   /  ---------- 
          /      2                /      2       
        \/    - b  + a*c        \/    - b  + a*c
then
the integral =
0 = 0
do backward replacement
0 = 0
Solution is:
       /       2        \
    log\c + a*x  + 2*b*x/
C + ---------------------
              2
The answer (Indefinite) [src] 
  /                                               
 |                              /   2            \
 |     a*x + b               log\a*x  + 2*b*x + c/
 | ---------------- dx = C + ---------------------
 |    2                                2          
 | a*x  + 2*b*x + c                               
 |                                                
/
$${{\log \left(a\,x^2+2\,b\,x+c\right)}\over{2}}$$
Simplify
The answer [src] 
log(a + c + 2*b)   log(c)
---------------- - ------
       2             2
$${{\log \left| c+2\,b+a\right| }\over{2}}-{{\log \left| c\right| }\over{2}}$$ 
=
= 
log(a + c + 2*b)   log(c)
---------------- - ------
       2             2
$$- \frac{\log{\left(c \right)}}{2} + \frac{\log{\left(a + 2 b + c \right)}}{2}$$
Simplify

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

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