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Resolución de integrales/ (x+4)/ (4+x)/ (scrt(4+x))/(x+4scrt(x+4))

Integral de (scrt(4+x))/(x+4scrt(x+4)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |       _______      
 |     \/ 4 + x       
 |  --------------- dx
 |          _______   
 |  x + 4*\/ x + 4    
 |                    
/                     
0
∫01x+4x+4x+4 dx\int\limits_{0}^{1} \frac{\sqrt{x + 4}}{x + 4 \sqrt{x + 4}}\, dx 
Integral(sqrt(4 + x)/(x + 4*sqrt(x + 4)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                                                    
 |                                                                                                                                     
 |      _______                             /         ___\                                  /         ___\                             
 |    \/ 4 + x                  _______     |     3*\/ 2 |    /      _______       ___\     |     3*\/ 2 |    /      _______       ___\
 | --------------- dx = C + 2*\/ 4 + x  + 2*|-2 - -------|*log\2 + \/ 4 + x  + 2*\/ 2 / + 2*|-2 + -------|*log\2 + \/ 4 + x  - 2*\/ 2 /
 |         _______                          \        2   /                                  \        2   /                             
 | x + 4*\/ x + 4                                                                                                                      
 |                                                                                                                                     
/
∫x+4x+4x+4 dx=C+2x+4+2(−322−2)log⁡(x+4+2+22)+2(−2+322)log⁡(x+4−22+2)\int \frac{\sqrt{x + 4}}{x + 4 \sqrt{x + 4}}\, dx = C + 2 \sqrt{x + 4} + 2 \left(- \frac{3 \sqrt{2}}{2} - 2\right) \log{\left(\sqrt{x + 4} + 2 + 2 \sqrt{2} \right)} + 2 \left(-2 + \frac{3 \sqrt{2}}{2}\right) \log{\left(\sqrt{x + 4} - 2 \sqrt{2} + 2 \right)}
Simplificar
Respuesta [src] 
                                                 1                                                                                                               
                                                 /                                                                                                               
                                                |                                                                                                                
                                                |  /                                          /   /                               2\                        2\   
                                                |  |               -3                         |   |                    /      ___\ |             /      ___\ |   
                                                |  |----------------------------------  for Or\And\x >= -4, x < -4 + 4*\1 - \/ 2 / /, x > -4 + 4*\1 - \/ 2 / /   
          /        ___\       ___               |  |  /                   2\                                                                                     
-4 - 4*log\1 + 4*\/ 5 / + 2*\/ 5  + 4*log(8) +  |  <  |    /      _______\ |                                                                                   dx
                                                |  |  |    \2 + \/ 4 + x / |   _______                                                                           
                                                |  |2*|1 - ----------------|*\/ 4 + x                                                                            
                                                |  |  \           8        /                                                                                     
                                                |  \                                                                                                             
                                                |                                                                                                                
                                               /                                                                                                                 
                                               0
−4log⁡(1+45)−4+∫01{−32(1−(x+4+2)28)x+4for (x≥−4∧x<−4+4(1−2)2)∨x>−4+4(1−2)2 dx+25+4log⁡(8)- 4 \log{\left(1 + 4 \sqrt{5} \right)} - 4 + \int\limits_{0}^{1} \begin{cases} - \frac{3}{2 \left(1 - \frac{\left(\sqrt{x + 4} + 2\right)^{2}}{8}\right) \sqrt{x + 4}} & \text{for}\: \left(x \geq -4 \wedge x < -4 + 4 \left(1 - \sqrt{2}\right)^{2}\right) \vee x > -4 + 4 \left(1 - \sqrt{2}\right)^{2} \end{cases}\, dx + 2 \sqrt{5} + 4 \log{\left(8 \right)} 
=
= 
                                                 1                                                                                                               
                                                 /                                                                                                               
                                                |                                                                                                                
                                                |  /                                          /   /                               2\                        2\   
                                                |  |               -3                         |   |                    /      ___\ |             /      ___\ |   
                                                |  |----------------------------------  for Or\And\x >= -4, x < -4 + 4*\1 - \/ 2 / /, x > -4 + 4*\1 - \/ 2 / /   
          /        ___\       ___               |  |  /                   2\                                                                                     
-4 - 4*log\1 + 4*\/ 5 / + 2*\/ 5  + 4*log(8) +  |  <  |    /      _______\ |                                                                                   dx
                                                |  |  |    \2 + \/ 4 + x / |   _______                                                                           
                                                |  |2*|1 - ----------------|*\/ 4 + x                                                                            
                                                |  |  \           8        /                                                                                     
                                                |  \                                                                                                             
                                                |                                                                                                                
                                               /                                                                                                                 
                                               0
−4log⁡(1+45)−4+∫01{−32(1−(x+4+2)28)x+4for (x≥−4∧x<−4+4(1−2)2)∨x>−4+4(1−2)2 dx+25+4log⁡(8)- 4 \log{\left(1 + 4 \sqrt{5} \right)} - 4 + \int\limits_{0}^{1} \begin{cases} - \frac{3}{2 \left(1 - \frac{\left(\sqrt{x + 4} + 2\right)^{2}}{8}\right) \sqrt{x + 4}} & \text{for}\: \left(x \geq -4 \wedge x < -4 + 4 \left(1 - \sqrt{2}\right)^{2}\right) \vee x > -4 + 4 \left(1 - \sqrt{2}\right)^{2} \end{cases}\, dx + 2 \sqrt{5} + 4 \log{\left(8 \right)} 
-4 - 4*log(1 + 4*sqrt(5)) + 2*sqrt(5) + 4*log(8) + Integral(Piecewise((-3/(2*(1 - (2 + sqrt(4 + x))^2/8)*sqrt(4 + x)), (x > -4 + 4*(1 - sqrt(2))^2)∨((x >= -4)∧(x < -4 + 4*(1 - sqrt(2))^2)))), (x, 0, 1))
Respuesta numérica [src] 
0.23653280823121
0.23653280823121

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

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