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Resolución de integrales/ x+1/x/ (x-2)/ (2+x)/ x+1/x2/(2+x)/(x-2)

Integral de x+1/x2/(2+x)/(x-2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
   ___                     
 \/ 3                      
   /                       
  |                        
  |   /    /    1     \\   
  |   |    |----------||   
  |   |    \x2*(2 + x)/|   
  |   |x + ------------| dx
  |   \       x - 2    /   
  |                        
 /                         
  ___                      
\/ 3                       
-----                      
  3
$$\int\limits_{\frac{\sqrt{3}}{3}}^{\sqrt{3}} \left(x + \frac{\frac{1}{x_{2}} \frac{1}{x + 2}}{x - 2}\right)\, dx$$ 
Integral(x + (1/(x2*(2 + x)))/(x - 2), (x, sqrt(3)/3, sqrt(3)))
Respuesta (Indefinida) [src] 
  /                                                         
 |                                                          
 | /    /    1     \\                                       
 | |    |----------||           2                           
 | |    \x2*(2 + x)/|          x    log(2 + x)   log(-2 + x)
 | |x + ------------| dx = C + -- - ---------- + -----------
 | \       x - 2    /          2       4*x2          4*x2   
 |                                                          
/
$$\int \left(x + \frac{\frac{1}{x_{2}} \frac{1}{x + 2}}{x - 2}\right)\, dx = C + \frac{x^{2}}{2} + \frac{\log{\left(x - 2 \right)}}{4 x_{2}} - \frac{\log{\left(x + 2 \right)}}{4 x_{2}}$$
Simplificar
Respuesta [src] 
                                                    /      ___\      /      ___\       
                                                    |    \/ 3 |      |    \/ 3 |       
         /      ___\      /      ___\            log|2 + -----|   log|2 - -----|       
      log\2 + \/ 3 /   log\2 - \/ 3 /   pi*I        \      3  /      \      3  /   pi*I
    - -------------- + -------------- + ----   - -------------- + -------------- + ----
4           4                4           4             4                4           4  
- + ---------------------------------------- - ----------------------------------------
3                      x2                                         x2
$$\frac{4}{3} - \frac{- \frac{\log{\left(\frac{\sqrt{3}}{3} + 2 \right)}}{4} + \frac{\log{\left(2 - \frac{\sqrt{3}}{3} \right)}}{4} + \frac{i \pi}{4}}{x_{2}} + \frac{- \frac{\log{\left(\sqrt{3} + 2 \right)}}{4} + \frac{\log{\left(2 - \sqrt{3} \right)}}{4} + \frac{i \pi}{4}}{x_{2}}$$ 
=
= 
                                                    /      ___\      /      ___\       
                                                    |    \/ 3 |      |    \/ 3 |       
         /      ___\      /      ___\            log|2 + -----|   log|2 - -----|       
      log\2 + \/ 3 /   log\2 - \/ 3 /   pi*I        \      3  /      \      3  /   pi*I
    - -------------- + -------------- + ----   - -------------- + -------------- + ----
4           4                4           4             4                4           4  
- + ---------------------------------------- - ----------------------------------------
3                      x2                                         x2
$$\frac{4}{3} - \frac{- \frac{\log{\left(\frac{\sqrt{3}}{3} + 2 \right)}}{4} + \frac{\log{\left(2 - \frac{\sqrt{3}}{3} \right)}}{4} + \frac{i \pi}{4}}{x_{2}} + \frac{- \frac{\log{\left(\sqrt{3} + 2 \right)}}{4} + \frac{\log{\left(2 - \sqrt{3} \right)}}{4} + \frac{i \pi}{4}}{x_{2}}$$ 
4/3 + (-log(2 + sqrt(3))/4 + log(2 - sqrt(3))/4 + pi*i/4)/x2 - (-log(2 + sqrt(3)/3)/4 + log(2 - sqrt(3)/3)/4 + pi*i/4)/x2

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

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