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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |     1       
 |  -------- dx
 |     2       
 |  - x  + 2   
 |             
/              
0
∫0112−x2 dx\int\limits_{0}^{1} \frac{1}{2 - x^{2}}\, dx 
Integral(1/(-x^2 + 2), (x, 0, 1))
Solución detallada

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

  1. Añadimos la constante de integración:

    {2acoth⁡(2x2)2for x2>22atanh⁡(2x2)2for x2<2+constant\begin{cases} \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} > 2 \\\frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} < 2 \end{cases}+ \mathrm{constant}

Respuesta:

{2acoth⁡(2x2)2for x2>22atanh⁡(2x2)2for x2<2+constant\begin{cases} \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} > 2 \\\frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} < 2 \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                     //           /    ___\            \
                     ||  ___      |x*\/ 2 |            |
                     ||\/ 2 *acoth|-------|            |
  /                  ||           \   2   /       2    |
 |                   ||--------------------  for x  > 2|
 |    1              ||         2                      |
 | -------- dx = C + |<                                |
 |    2              ||           /    ___\            |
 | - x  + 2          ||  ___      |x*\/ 2 |            |
 |                   ||\/ 2 *atanh|-------|            |
/                    ||           \   2   /       2    |
                     ||--------------------  for x  < 2|
                     \\         2                      /
∫12−x2 dx=C+{2acoth⁡(2x2)2for x2>22atanh⁡(2x2)2for x2<2\int \frac{1}{2 - x^{2}}\, dx = C + \begin{cases} \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} > 2 \\\frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} < 2 \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.01.5
Trazar el gráfico f(x)
Respuesta [src] 
    ___ /          /       ___\\     ___    /  ___\     ___ /          /  ___\\     ___    /      ___\
  \/ 2 *\pi*I + log\-1 + \/ 2 //   \/ 2 *log\\/ 2 /   \/ 2 *\pi*I + log\\/ 2 //   \/ 2 *log\1 + \/ 2 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
                4                         4                       4                        4
−2log⁡(2)4+2log⁡(1+2)4−2(log⁡(−1+2)+iπ)4+2(log⁡(2)+iπ)4- \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{4} - \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{4} 
=
= 
    ___ /          /       ___\\     ___    /  ___\     ___ /          /  ___\\     ___    /      ___\
  \/ 2 *\pi*I + log\-1 + \/ 2 //   \/ 2 *log\\/ 2 /   \/ 2 *\pi*I + log\\/ 2 //   \/ 2 *log\1 + \/ 2 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
                4                         4                       4                        4
−2log⁡(2)4+2log⁡(1+2)4−2(log⁡(−1+2)+iπ)4+2(log⁡(2)+iπ)4- \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{4} - \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{4} 
-sqrt(2)*(pi*i + log(-1 + sqrt(2)))/4 - sqrt(2)*log(sqrt(2))/4 + sqrt(2)*(pi*i + log(sqrt(2)))/4 + sqrt(2)*log(1 + 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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