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Resolución de integrales/ (x+2)/ √(x+2)/(x-2√(x+2))

Integral de √(x+2)/(x-2√(x+2)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |       _______      
 |     \/ x + 2       
 |  --------------- dx
 |          _______   
 |  x - 2*\/ x + 2    
 |                    
/                     
0
∫01x+2x−2x+2 dx\int\limits_{0}^{1} \frac{\sqrt{x + 2}}{x - 2 \sqrt{x + 2}}\, dx 
Integral(sqrt(x + 2)/(x - 2*sqrt(x + 2)), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                                     //            /  ___ /       _______\\                            \
                                                                     ||   ___      |\/ 3 *\-1 + \/ 2 + x /|                            |
  /                                                                  ||-\/ 3 *acoth|----------------------|                       2    |
 |                                                                   ||            \          3           /       /       _______\     |
 |      _______                                                      ||-------------------------------------  for \-1 + \/ 2 + x /  > 3|
 |    \/ x + 2                  _______        /        _______\     ||                  3                                             |
 | --------------- dx = C + 2*\/ 2 + x  + 2*log\x - 2*\/ 2 + x / + 8*|<                                                                |
 |         _______                                                   ||            /  ___ /       _______\\                            |
 | x - 2*\/ x + 2                                                    ||   ___      |\/ 3 *\-1 + \/ 2 + x /|                            |
 |                                                                   ||-\/ 3 *atanh|----------------------|                       2    |
/                                                                    ||            \          3           /       /       _______\     |
                                                                     ||-------------------------------------  for \-1 + \/ 2 + x /  < 3|
                                                                     \\                  3                                             /
∫x+2x−2x+2 dx=C+2x+2+8({−3acoth⁡(3(x+2−1)3)3for (x+2−1)2>3−3atanh⁡(3(x+2−1)3)3for (x+2−1)2<3)+2log⁡(x−2x+2)\int \frac{\sqrt{x + 2}}{x - 2 \sqrt{x + 2}}\, dx = C + 2 \sqrt{x + 2} + 8 \left(\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} \left(\sqrt{x + 2} - 1\right)}{3} \right)}}{3} & \text{for}\: \left(\sqrt{x + 2} - 1\right)^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} \left(\sqrt{x + 2} - 1\right)}{3} \right)}}{3} & \text{for}\: \left(\sqrt{x + 2} - 1\right)^{2} < 3 \end{cases}\right) + 2 \log{\left(x - 2 \sqrt{x + 2} \right)}
Simplificar
Respuesta [src] 
                                                               1                                                                                                            
                                                               /                                                                                                            
                                                              |                                                                                                             
                                                              |  /                                           /   /                             2\                      2\   
                                                              |  |                -4                         |   |                  /      ___\ |           /      ___\ |   
                                                              |  |-----------------------------------  for Or\And\x >= -2, x < -2 + \1 + \/ 3 / /, x > -2 + \1 + \/ 3 / /   
      ___        /    ___\       ___        /         ___\    |  |  /                    2\                                                                                 
- 2*\/ 2  - 2*log\2*\/ 2 / + 2*\/ 3  + 2*log\-1 + 2*\/ 3 / +  |  <  |    /       _______\ |                                                                               dx
                                                              |  |  |    \-1 + \/ 2 + x / |   _______                                                                       
                                                              |  |3*|1 - -----------------|*\/ 2 + x                                                                        
                                                              |  |  \            3        /                                                                                 
                                                              |  \                                                                                                          
                                                              |                                                                                                             
                                                             /                                                                                                              
                                                             0
−22−2log⁡(22)+∫01{−43(1−(x+2−1)23)x+2for (x≥−2∧x<−2+(1+3)2)∨x>−2+(1+3)2 dx+2log⁡(−1+23)+23- 2 \sqrt{2} - 2 \log{\left(2 \sqrt{2} \right)} + \int\limits_{0}^{1} \begin{cases} - \frac{4}{3 \left(1 - \frac{\left(\sqrt{x + 2} - 1\right)^{2}}{3}\right) \sqrt{x + 2}} & \text{for}\: \left(x \geq -2 \wedge x < -2 + \left(1 + \sqrt{3}\right)^{2}\right) \vee x > -2 + \left(1 + \sqrt{3}\right)^{2} \end{cases}\, dx + 2 \log{\left(-1 + 2 \sqrt{3} \right)} + 2 \sqrt{3} 
=
= 
                                                               1                                                                                                            
                                                               /                                                                                                            
                                                              |                                                                                                             
                                                              |  /                                           /   /                             2\                      2\   
                                                              |  |                -4                         |   |                  /      ___\ |           /      ___\ |   
                                                              |  |-----------------------------------  for Or\And\x >= -2, x < -2 + \1 + \/ 3 / /, x > -2 + \1 + \/ 3 / /   
      ___        /    ___\       ___        /         ___\    |  |  /                    2\                                                                                 
- 2*\/ 2  - 2*log\2*\/ 2 / + 2*\/ 3  + 2*log\-1 + 2*\/ 3 / +  |  <  |    /       _______\ |                                                                               dx
                                                              |  |  |    \-1 + \/ 2 + x / |   _______                                                                       
                                                              |  |3*|1 - -----------------|*\/ 2 + x                                                                        
                                                              |  |  \            3        /                                                                                 
                                                              |  \                                                                                                          
                                                              |                                                                                                             
                                                             /                                                                                                              
                                                             0
−22−2log⁡(22)+∫01{−43(1−(x+2−1)23)x+2for (x≥−2∧x<−2+(1+3)2)∨x>−2+(1+3)2 dx+2log⁡(−1+23)+23- 2 \sqrt{2} - 2 \log{\left(2 \sqrt{2} \right)} + \int\limits_{0}^{1} \begin{cases} - \frac{4}{3 \left(1 - \frac{\left(\sqrt{x + 2} - 1\right)^{2}}{3}\right) \sqrt{x + 2}} & \text{for}\: \left(x \geq -2 \wedge x < -2 + \left(1 + \sqrt{3}\right)^{2}\right) \vee x > -2 + \left(1 + \sqrt{3}\right)^{2} \end{cases}\, dx + 2 \log{\left(-1 + 2 \sqrt{3} \right)} + 2 \sqrt{3} 
-2*sqrt(2) - 2*log(2*sqrt(2)) + 2*sqrt(3) + 2*log(-1 + 2*sqrt(3)) + Integral(Piecewise((-4/(3*(1 - (-1 + sqrt(2 + x))^2/3)*sqrt(2 + x)), (x > -2 + (1 + sqrt(3))^2)∨((x >= -2)∧(x < -2 + (1 + sqrt(3))^2)))), (x, 0, 1))
Respuesta numérica [src] 
-0.596413293796112
-0.596413293796112

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

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