Sr Examen

Otras calculadoras

Integral de (x+4)/(2x²-6x+8) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1                  
  /                  
 |                   
 |      x + 4        
 |  -------------- dx
 |     2             
 |  2*x  - 6*x + 8   
 |                   
/                    
0                    
$$\int\limits_{0}^{1} \frac{x + 4}{\left(2 x^{2} - 6 x\right) + 8}\, dx$$
Integral((x + 4)/(2*x^2 - 6*x + 8), (x, 0, 1))
Solución detallada
Tenemos el integral:
  /                 
 |                  
 |     x + 4        
 | -------------- dx
 |    2             
 | 2*x  - 6*x + 8   
 |                  
/                   
Reescribimos la función subintegral
                 /  2*2*x - 6   \                              
                 |--------------|             /  11 \          
                 |   2          |             |-----|          
    x + 4        \2*x  - 6*x + 8/             \2*7/2/          
-------------- = ---------------- + ---------------------------
   2                    4                                 2    
2*x  - 6*x + 8                      /     ___         ___\     
                                    |-2*\/ 7      3*\/ 7 |     
                                    |--------*x + -------|  + 1
                                    \   7            7   /     
o
  /                   
 |                    
 |     x + 4          
 | -------------- dx  
 |    2              =
 | 2*x  - 6*x + 8     
 |                    
/                     
  
                            /                              
                           |                               
                           |              1                
  /                    11* | --------------------------- dx
 |                         |                       2       
 |   2*2*x - 6             | /     ___         ___\        
 | -------------- dx       | |-2*\/ 7      3*\/ 7 |        
 |    2                    | |--------*x + -------|  + 1   
 | 2*x  - 6*x + 8          | \   7            7   /        
 |                         |                               
/                         /                                
-------------------- + ------------------------------------
         4                              7                  
En integral
  /                 
 |                  
 |   2*2*x - 6      
 | -------------- dx
 |    2             
 | 2*x  - 6*x + 8   
 |                  
/                   
--------------------
         4          
hacemos el cambio
              2
u = -6*x + 2*x 
entonces
integral =
  /                     
 |                      
 |   1                  
 | ----- du             
 | 8 + u                
 |                      
/             log(8 + u)
----------- = ----------
     4            4     
hacemos cambio inverso
  /                                     
 |                                      
 |   2*2*x - 6                          
 | -------------- dx                    
 |    2                                 
 | 2*x  - 6*x + 8                       
 |                        /     2      \
/                      log\4 + x  - 3*x/
-------------------- = -----------------
         4                     4        
En integral
     /                              
    |                               
    |              1                
11* | --------------------------- dx
    |                       2       
    | /     ___         ___\        
    | |-2*\/ 7      3*\/ 7 |        
    | |--------*x + -------|  + 1   
    | \   7            7   /        
    |                               
   /                                
------------------------------------
                 7                  
hacemos el cambio
        ___         ___
    3*\/ 7    2*x*\/ 7 
v = ------- - ---------
       7          7    
entonces
integral =
     /                      
    |                       
    |   1                   
11* | ------ dv             
    |      2                
    | 1 + v                 
    |                       
   /              11*atan(v)
--------------- = ----------
       7              7     
hacemos cambio inverso
     /                                                                     
    |                                                                      
    |              1                                                       
11* | --------------------------- dx                                       
    |                       2                                              
    | /     ___         ___\                                               
    | |-2*\/ 7      3*\/ 7 |                                               
    | |--------*x + -------|  + 1                   /      ___         ___\
    | \   7            7   /                ___     |  3*\/ 7    2*x*\/ 7 |
    |                                  11*\/ 7 *atan|- ------- + ---------|
   /                                                \     7          7    /
------------------------------------ = ------------------------------------
                 7                                      14                 
La solución:
                                     /      ___         ___\
                             ___     |  3*\/ 7    2*x*\/ 7 |
       /     2      \   11*\/ 7 *atan|- ------- + ---------|
    log\4 + x  - 3*x/                \     7          7    /
C + ----------------- + ------------------------------------
            4                            14                 
Respuesta (Indefinida) [src]
                                                            /    ___           \
  /                                                 ___     |2*\/ 7 *(-3/2 + x)|
 |                            /     2      \   11*\/ 7 *atan|------------------|
 |     x + 4               log\4 + x  - 3*x/                \        7         /
 | -------------- dx = C + ----------------- + ---------------------------------
 |    2                            4                           14               
 | 2*x  - 6*x + 8                                                               
 |                                                                              
/                                                                               
$$\int \frac{x + 4}{\left(2 x^{2} - 6 x\right) + 8}\, dx = C + \frac{\log{\left(x^{2} - 3 x + 4 \right)}}{4} + \frac{11 \sqrt{7} \operatorname{atan}{\left(\frac{2 \sqrt{7} \left(x - \frac{3}{2}\right)}{7} \right)}}{14}$$
Gráfica
Respuesta [src]
                                 /  ___\                /    ___\
                         ___     |\/ 7 |        ___     |3*\/ 7 |
                    11*\/ 7 *atan|-----|   11*\/ 7 *atan|-------|
  log(4)   log(2)                \  7  /                \   7   /
- ------ + ------ - -------------------- + ----------------------
    4        4               14                      14          
$$- \frac{11 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{14} - \frac{\log{\left(4 \right)}}{4} + \frac{\log{\left(2 \right)}}{4} + \frac{11 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{14}$$
=
=
                                 /  ___\                /    ___\
                         ___     |\/ 7 |        ___     |3*\/ 7 |
                    11*\/ 7 *atan|-----|   11*\/ 7 *atan|-------|
  log(4)   log(2)                \  7  /                \   7   /
- ------ + ------ - -------------------- + ----------------------
    4        4               14                      14          
$$- \frac{11 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{14} - \frac{\log{\left(4 \right)}}{4} + \frac{\log{\left(2 \right)}}{4} + \frac{11 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{14}$$
-log(4)/4 + log(2)/4 - 11*sqrt(7)*atan(sqrt(7)/7)/14 + 11*sqrt(7)*atan(3*sqrt(7)/7)/14
Respuesta numérica [src]
0.838456917020988
0.838456917020988

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.