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(3x-2)/(x^2+9)

Integral of (3x-2)/(x^2+9) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |  3*x - 2   
 |  ------- dx
 |    2       
 |   x  + 9   
 |            
/             
0             
$$\int\limits_{0}^{1} \frac{3 x - 2}{x^{2} + 9}\, dx$$
Integral((3*x - 1*2)/(x^2 + 9), (x, 0, 1))
Detail solution
We have the integral:
  /            
 |             
 |   3*x - 2   
 | 1*------- dx
 |     2       
 |    x  + 9   
 |             
/              
Rewrite the integrand
              1*2*x + 0                    
          3*--------------       /-2 \     
               2                 |---|     
3*x - 2     1*x  + 0*x + 9       \ 9 /     
------- = ---------------- + --------------
  2              2                    2    
 x  + 9                      /  x    \     
                             |- - + 0|  + 1
                             \  3    /     
or
  /              
 |               
 |   3*x - 2     
 | 1*------- dx  
 |     2        =
 |    x  + 9     
 |               
/                
  
      /                                          
     |                                           
     |       1                 /                 
  2* | -------------- dx      |                  
     |          2             |   1*2*x + 0      
     | /  x    \           3* | -------------- dx
     | |- - + 0|  + 1         |    2             
     | \  3    /              | 1*x  + 0*x + 9   
     |                        |                  
    /                        /                   
- ---------------------- + ----------------------
            9                        2           
In the integral
    /                 
   |                  
   |   1*2*x + 0      
3* | -------------- dx
   |    2             
   | 1*x  + 0*x + 9   
   |                  
  /                   
----------------------
          2           
do replacement
     2
u = x 
then
the integral =
    /                       
   |                        
   |   1                    
3* | ----- du               
   | 9 + u                  
   |                        
  /             3*log(9 + u)
------------- = ------------
      2              2      
do backward replacement
    /                                 
   |                                  
   |   1*2*x + 0                      
3* | -------------- dx                
   |    2                             
   | 1*x  + 0*x + 9                   
   |                          /     2\
  /                      3*log\9 + x /
---------------------- = -------------
          2                    2      
In the integral
     /                 
    |                  
    |       1          
-2* | -------------- dx
    |          2       
    | /  x    \        
    | |- - + 0|  + 1   
    | \  3    /        
    |                  
   /                   
-----------------------
           9           
do replacement
    -x 
v = ---
     3 
then
the integral =
     /                      
    |                       
    |   1                   
-2* | ------ dv             
    |      2                
    | 1 + v                 
    |                       
   /              -2*atan(v)
--------------- = ----------
       9              9     
do backward replacement
     /                              
    |                               
    |       1                       
-2* | -------------- dx             
    |          2                    
    | /  x    \                     
    | |- - + 0|  + 1                
    | \  3    /                  /x\
    |                     -2*atan|-|
   /                             \3/
----------------------- = ----------
           9                  3     
Solution is:
          /x\                
    2*atan|-|        /     2\
          \3/   3*log\9 + x /
C - --------- + -------------
        3             2      
The answer (Indefinite) [src]
  /                       /x\                
 |                  2*atan|-|        /     2\
 | 3*x - 2                \3/   3*log\9 + x /
 | ------- dx = C - --------- + -------------
 |   2                  3             2      
 |  x  + 9                                   
 |                                           
/                                            
$${{3\,\log \left(x^2+9\right)}\over{2}}-{{2\,\arctan \left({{x }\over{3}}\right)}\over{3}}$$
The graph
The answer [src]
  3*log(9)   2*atan(1/3)   3*log(10)
- -------- - ----------- + ---------
     2            3            2    
$${{3\,\log 10}\over{2}}-{{3\,\log 9}\over{2}}-{{2\,\arctan \left({{1 }\over{3}}\right)}\over{3}}$$
=
=
  3*log(9)   2*atan(1/3)   3*log(10)
- -------- - ----------- + ---------
     2            3            2    
$$- \frac{3 \log{\left(9 \right)}}{2} - \frac{2 \operatorname{atan}{\left(\frac{1}{3} \right)}}{3} + \frac{3 \log{\left(10 \right)}}{2}$$
Numerical answer [src]
-0.056459596111022
-0.056459596111022
The graph
Integral of (3x-2)/(x^2+9) dx

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