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Resolución de integrales/ dy/(y^2-2)

Integral de dy/(y^2-2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1          
  /          
 |           
 |    1      
 |  ------ dy
 |   2       
 |  y  - 2   
 |           
/            
0
∫011y2−2 dy\int\limits_{0}^{1} \frac{1}{y^{2} - 2}\, dy 
Integral(1/(y^2 - 2), (y, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=1, c=-2, context=1/(y**2 - 2), symbol=y), False), (ArccothRule(a=1, b=1, c=-2, context=1/(y**2 - 2), symbol=y), y**2 > 2), (ArctanhRule(a=1, b=1, c=-2, context=1/(y**2 - 2), symbol=y), y**2 < 2)], context=1/(y**2 - 2), symbol=y)

  1. Añadimos la constante de integración:

    {−2acoth⁡(2y2)2for y2>2−2atanh⁡(2y2)2for y2<2+constant\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} y}{2} \right)}}{2} & \text{for}\: y^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} y}{2} \right)}}{2} & \text{for}\: y^{2} < 2 \end{cases}+ \mathrm{constant}

Respuesta:

{−2acoth⁡(2y2)2for y2>2−2atanh⁡(2y2)2for y2<2+constant\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} y}{2} \right)}}{2} & \text{for}\: y^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} y}{2} \right)}}{2} & \text{for}\: y^{2} < 2 \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                   //            /    ___\             \
                   ||   ___      |y*\/ 2 |             |
                   ||-\/ 2 *acoth|-------|             |
  /                ||            \   2   /        2    |
 |                 ||----------------------  for y  > 2|
 |   1             ||          2                       |
 | ------ dy = C + |<                                  |
 |  2              ||            /    ___\             |
 | y  - 2          ||   ___      |y*\/ 2 |             |
 |                 ||-\/ 2 *atanh|-------|             |
/                  ||            \   2   /        2    |
                   ||----------------------  for y  < 2|
                   \\          2                       /
∫1y2−2 dy=C+{−2acoth⁡(2y2)2for y2>2−2atanh⁡(2y2)2for y2<2\int \frac{1}{y^{2} - 2}\, dy = C + \begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} y}{2} \right)}}{2} & \text{for}\: y^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} y}{2} \right)}}{2} & \text{for}\: y^{2} < 2 \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-1.50.0
Trazar el gráfico f(x)
Respuesta [src] 
    ___ /          /  ___\\     ___    /      ___\     ___ /          /       ___\\     ___    /  ___\
  \/ 2 *\pi*I + log\\/ 2 //   \/ 2 *log\1 + \/ 2 /   \/ 2 *\pi*I + log\-1 + \/ 2 //   \/ 2 *log\\/ 2 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
              4                        4                           4                         4
−2log⁡(1+2)4+2log⁡(2)4−2(log⁡(2)+iπ)4+2(log⁡(−1+2)+iπ)4- \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{4} - \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{4} 
=
= 
    ___ /          /  ___\\     ___    /      ___\     ___ /          /       ___\\     ___    /  ___\
  \/ 2 *\pi*I + log\\/ 2 //   \/ 2 *log\1 + \/ 2 /   \/ 2 *\pi*I + log\-1 + \/ 2 //   \/ 2 *log\\/ 2 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
              4                        4                           4                         4
−2log⁡(1+2)4+2log⁡(2)4−2(log⁡(2)+iπ)4+2(log⁡(−1+2)+iπ)4- \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{4} - \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{4} 
-sqrt(2)*(pi*i + log(sqrt(2)))/4 - sqrt(2)*log(1 + sqrt(2))/4 + sqrt(2)*(pi*i + log(-1 + sqrt(2)))/4 + sqrt(2)*log(sqrt(2))/4
Respuesta numérica [src] 
-0.623225240140231
-0.623225240140231

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

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