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Resolución de integrales/ 1+2/x/ xroot|1+2/x|

Integral de xroot|1+2/x| dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |        _________   
 |       / |    2|    
 |  x*  /  |1 + -|  dx
 |    \/   |    x|    
 |                    
/                     
0
∫01x∣1+2x∣ dx\int\limits_{0}^{1} x \sqrt{\left|{1 + \frac{2}{x}}\right|}\, dx 
Integral(x*sqrt(|1 + 2/x|), (x, 0, 1))
Respuesta (Indefinida) [src] 
                            //                                      /        _______\                                                                                                                               _______                     _______              \
                            ||                               ___    |       /     2 |                                                                    ___    /1\                                         ___    /     2           ___  2    /     2               |
                            ||              ___         pi*\/ 2 *log|1 +   /  1 + - |               ___  2                      ___  2              pi*\/ 2 *log|-|                    ___           pi*x*\/ 2 *  /  1 + -      pi*\/ 2 *x *  /  1 + -               |
                            ||       pi*x*\/ 2                      \    \/       x /          pi*\/ 2 *x                  pi*\/ 2 *x                           \x/             pi*x*\/ 2                       \/       x                  \/       x         2     |
                            ||----------------------- + ----------------------------- + ------------------------- + ------------------------- + ----------------------- - ---------------------- - ------------------------- - -------------------------  for --- < 1|
  /                         ||Gamma(-9/4)*Gamma(13/4)      Gamma(-9/4)*Gamma(13/4)      2*Gamma(-9/4)*Gamma(13/4)   2*Gamma(-7/4)*Gamma(11/4)   2*Gamma(1/4)*Gamma(3/4)   Gamma(-3/4)*Gamma(7/4)   2*Gamma(-9/4)*Gamma(13/4)   2*Gamma(-9/4)*Gamma(13/4)      |x|    |
 |                          ||                                                                                                                                                                                                                                       |
 |       _________          ||                                                         /  ___   ___\                                                                                                                                                                 |
 |      / |    2|           ||                                                ___      |\/ 2 *\/ x |                                                                                                                                                                 |
 | x*  /  |1 + -|  dx = C + |<                                         pi*I*\/ 2 *asinh|-----------|                  ___   ___                           ___  3/2                           ___  5/2                                                                |
 |   \/   |    x|           ||                                                         \     2     /           pi*I*\/ 2 *\/ x                   3*pi*I*\/ 2 *x                       pi*I*\/ 2 *x                                                            |x|    |
 |                          ||                                         ----------------------------- - ------------------------------- - --------------------------------- - ---------------------------------                                            for --- < 1|
/                           ||                                             Gamma(1/4)*Gamma(3/4)         _______                             _______                             _______                                                                       2     |
                            ||                                                                         \/ 2 + x *Gamma(1/4)*Gamma(3/4)   2*\/ 2 + x *Gamma(1/4)*Gamma(3/4)   2*\/ 2 + x *Gamma(1/4)*Gamma(3/4)                                                       |
                            ||                                                                                                                                                                                                                                       |
                            ||                                                                                                           /               |    -pi*I\                                                                                                 |
                            ||                                                                                        ___   ____  __2, 1 |-1/2   -5/4, 1 | 2*e     |                                                                                                 |
                            ||                                                                                   -2*\/ 2 *\/ pi */__     |               | --------|                                                                                       otherwise |
                            \\                                                                                                   \_|3, 3 \-2, 0   -5/4   |    x    /                                                                                                 /
∫x∣1+2x∣ dx=C+{−2πx21+2x2Γ(−94)Γ(134)+2πx22Γ(−94)Γ(134)+2πx22Γ(−74)Γ(114)−2πx1+2x2Γ(−94)Γ(134)+2πxΓ(−94)Γ(134)−2πxΓ(−34)Γ(74)+2πlog⁡(1x)2Γ(14)Γ(34)+2πlog⁡(1+2x+1)Γ(−94)Γ(134)for 2∣x∣<1−2iπx522x+2Γ(14)Γ(34)−32iπx322x+2Γ(14)Γ(34)−2iπxx+2Γ(14)Γ(34)+2iπasinh⁡(2x2)Γ(14)Γ(34)for ∣x∣2<1−22πG3,32,1(−12−54,1−2,0−54|2e−iπx)otherwise\int x \sqrt{\left|{1 + \frac{2}{x}}\right|}\, dx = C + \begin{cases} - \frac{\sqrt{2} \pi x^{2} \sqrt{1 + \frac{2}{x}}}{2 \Gamma\left(- \frac{9}{4}\right) \Gamma\left(\frac{13}{4}\right)} + \frac{\sqrt{2} \pi x^{2}}{2 \Gamma\left(- \frac{9}{4}\right) \Gamma\left(\frac{13}{4}\right)} + \frac{\sqrt{2} \pi x^{2}}{2 \Gamma\left(- \frac{7}{4}\right) \Gamma\left(\frac{11}{4}\right)} - \frac{\sqrt{2} \pi x \sqrt{1 + \frac{2}{x}}}{2 \Gamma\left(- \frac{9}{4}\right) \Gamma\left(\frac{13}{4}\right)} + \frac{\sqrt{2} \pi x}{\Gamma\left(- \frac{9}{4}\right) \Gamma\left(\frac{13}{4}\right)} - \frac{\sqrt{2} \pi x}{\Gamma\left(- \frac{3}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{2} \pi \log{\left(\frac{1}{x} \right)}}{2 \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} + \frac{\sqrt{2} \pi \log{\left(\sqrt{1 + \frac{2}{x}} + 1 \right)}}{\Gamma\left(- \frac{9}{4}\right) \Gamma\left(\frac{13}{4}\right)} & \text{for}\: \frac{2}{\left|{x}\right|} < 1 \\- \frac{\sqrt{2} i \pi x^{\frac{5}{2}}}{2 \sqrt{x + 2} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} - \frac{3 \sqrt{2} i \pi x^{\frac{3}{2}}}{2 \sqrt{x + 2} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} - \frac{\sqrt{2} i \pi \sqrt{x}}{\sqrt{x + 2} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} + \frac{\sqrt{2} i \pi \operatorname{asinh}{\left(\frac{\sqrt{2} \sqrt{x}}{2} \right)}}{\Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} & \text{for}\: \frac{\left|{x}\right|}{2} < 1 \\- 2 \sqrt{2} \sqrt{\pi} {G_{3, 3}^{2, 1}\left(\begin{matrix} - \frac{1}{2} & - \frac{5}{4}, 1 \\-2, 0 & - \frac{5}{4} \end{matrix} \middle| {\frac{2 e^{- i \pi}}{x}} \right)} & \text{otherwise} \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.9002
Trazar el gráfico f(x)
Respuesta numérica [src] 
1.07357185910647
1.07357185910647

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

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