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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |        _________   
 |  x + \/ 2*x + 1    
 |  --------------- dx
 |        _________   
 |  x - \/ 2*x + 1    
 |                    
/                     
0
∫01x+2x+1x−2x+1 dx\int\limits_{0}^{1} \frac{x + \sqrt{2 x + 1}}{x - \sqrt{2 x + 1}}\, dx 
Integral((x + sqrt(2*x + 1))/(x - sqrt(2*x + 1)), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                                                      //            /  ___ /       _________\\                              \
                                                                                      ||   ___      |\/ 2 *\-1 + \/ 2*x + 1 /|                              |
  /                                                                                   ||-\/ 2 *acoth|------------------------|                         2    |
 |                                                                                    ||            \           2            /       /       _________\     |
 |       _________                                                                    ||---------------------------------------  for \-1 + \/ 2*x + 1 /  > 2|
 | x + \/ 2*x + 1       1               _________        /      _________      \      ||                   2                                                |
 | --------------- dx = - + C + x + 4*\/ 2*x + 1  + 4*log\- 2*\/ 2*x + 1  + 2*x/ + 12*|<                                                                    |
 |       _________      2                                                             ||            /  ___ /       _________\\                              |
 | x - \/ 2*x + 1                                                                     ||   ___      |\/ 2 *\-1 + \/ 2*x + 1 /|                              |
 |                                                                                    ||-\/ 2 *atanh|------------------------|                         2    |
/                                                                                     ||            \           2            /       /       _________\     |
                                                                                      ||---------------------------------------  for \-1 + \/ 2*x + 1 /  < 2|
                                                                                      \\                   2                                                /
∫x+2x+1x−2x+1 dx=C+x+42x+1+12({−2acoth⁡(2(2x+1−1)2)2for (2x+1−1)2>2−2atanh⁡(2(2x+1−1)2)2for (2x+1−1)2<2)+4log⁡(2x−22x+1)+12\int \frac{x + \sqrt{2 x + 1}}{x - \sqrt{2 x + 1}}\, dx = C + x + 4 \sqrt{2 x + 1} + 12 \left(\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} \left(\sqrt{2 x + 1} - 1\right)}{2} \right)}}{2} & \text{for}\: \left(\sqrt{2 x + 1} - 1\right)^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} \left(\sqrt{2 x + 1} - 1\right)}{2} \right)}}{2} & \text{for}\: \left(\sqrt{2 x + 1} - 1\right)^{2} < 2 \end{cases}\right) + 4 \log{\left(2 x - 2 \sqrt{2 x + 1} \right)} + \frac{1}{2}
Simplificar
Respuesta [src] 
                                                  1                                                                                                                  
                                                  /                                                                                                                  
                                                 |                                                                                                                   
                                                 |  /                                             /   /                                2\                       2\   
                                                 |  |                                             |   |                     /      ___\ |            /      ___\ |   
                                                 |  |                 -6                          |   |                 1   \1 + \/ 2 / |        1   \1 + \/ 2 / |   
                    ___        /         ___\    |  |-------------------------------------  for Or|And|x >= -1/2, x < - - + ------------|, x > - - + ------------|   
-3 - 4*log(2) + 4*\/ 3  + 4*log\-2 + 2*\/ 3 / +  |  <            /                      2\        \   \                 2        2      /        2        2      / dx
                                                 |  |            |    /       _________\ |                                                                           
                                                 |  |  _________ |    \-1 + \/ 1 + 2*x / |                                                                           
                                                 |  |\/ 1 + 2*x *|1 - -------------------|                                                                           
                                                 |  \            \             2         /                                                                           
                                                 |                                                                                                                   
                                                /                                                                                                                    
                                                0
∫01{−6(1−(2x+1−1)22)2x+1for (x≥−12∧x<−12+(1+2)22)∨x>−12+(1+2)22 dx−3−4log⁡(2)+4log⁡(−2+23)+43\int\limits_{0}^{1} \begin{cases} - \frac{6}{\left(1 - \frac{\left(\sqrt{2 x + 1} - 1\right)^{2}}{2}\right) \sqrt{2 x + 1}} & \text{for}\: \left(x \geq - \frac{1}{2} \wedge x < - \frac{1}{2} + \frac{\left(1 + \sqrt{2}\right)^{2}}{2}\right) \vee x > - \frac{1}{2} + \frac{\left(1 + \sqrt{2}\right)^{2}}{2} \end{cases}\, dx - 3 - 4 \log{\left(2 \right)} + 4 \log{\left(-2 + 2 \sqrt{3} \right)} + 4 \sqrt{3} 
=
= 
                                                  1                                                                                                                  
                                                  /                                                                                                                  
                                                 |                                                                                                                   
                                                 |  /                                             /   /                                2\                       2\   
                                                 |  |                                             |   |                     /      ___\ |            /      ___\ |   
                                                 |  |                 -6                          |   |                 1   \1 + \/ 2 / |        1   \1 + \/ 2 / |   
                    ___        /         ___\    |  |-------------------------------------  for Or|And|x >= -1/2, x < - - + ------------|, x > - - + ------------|   
-3 - 4*log(2) + 4*\/ 3  + 4*log\-2 + 2*\/ 3 / +  |  <            /                      2\        \   \                 2        2      /        2        2      / dx
                                                 |  |            |    /       _________\ |                                                                           
                                                 |  |  _________ |    \-1 + \/ 1 + 2*x / |                                                                           
                                                 |  |\/ 1 + 2*x *|1 - -------------------|                                                                           
                                                 |  \            \             2         /                                                                           
                                                 |                                                                                                                   
                                                /                                                                                                                    
                                                0
∫01{−6(1−(2x+1−1)22)2x+1for (x≥−12∧x<−12+(1+2)22)∨x>−12+(1+2)22 dx−3−4log⁡(2)+4log⁡(−2+23)+43\int\limits_{0}^{1} \begin{cases} - \frac{6}{\left(1 - \frac{\left(\sqrt{2 x + 1} - 1\right)^{2}}{2}\right) \sqrt{2 x + 1}} & \text{for}\: \left(x \geq - \frac{1}{2} \wedge x < - \frac{1}{2} + \frac{\left(1 + \sqrt{2}\right)^{2}}{2}\right) \vee x > - \frac{1}{2} + \frac{\left(1 + \sqrt{2}\right)^{2}}{2} \end{cases}\, dx - 3 - 4 \log{\left(2 \right)} + 4 \log{\left(-2 + 2 \sqrt{3} \right)} + 4 \sqrt{3} 
-3 - 4*log(2) + 4*sqrt(3) + 4*log(-2 + 2*sqrt(3)) + Integral(Piecewise((-6/(sqrt(1 + 2*x)*(1 - (-1 + sqrt(1 + 2*x))^2/2)), (x > -1/2 + (1 + sqrt(2))^2/2)∨((x >= -1/2)∧(x < -1/2 + (1 + sqrt(2))^2/2)))), (x, 0, 1))
Respuesta numérica [src] 
-2.18240013931476
-2.18240013931476

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

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