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Resolución de integrales/ 8-x^2/ 1/(8-x^2)

Integral de 1/(8-x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1          
  /          
 |           
 |    1      
 |  ------ dx
 |       2   
 |  8 - x    
 |           
/            
0
∫0118−x2 dx\int\limits_{0}^{1} \frac{1}{8 - x^{2}}\, dx 
Integral(1/(8 - x^2), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-1, c=8, context=1/(8 - x**2), symbol=x), False), (ArccothRule(a=1, b=-1, c=8, context=1/(8 - x**2), symbol=x), x**2 > 8), (ArctanhRule(a=1, b=-1, c=8, context=1/(8 - x**2), symbol=x), x**2 < 8)], context=1/(8 - x**2), symbol=x)

  1. Añadimos la constante de integración:

    {2acoth⁡(2x4)4for x2>82atanh⁡(2x4)4for x2<8+constant\begin{cases} \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{4} \right)}}{4} & \text{for}\: x^{2} > 8 \\\frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{4} \right)}}{4} & \text{for}\: x^{2} < 8 \end{cases}+ \mathrm{constant}

Respuesta:

{2acoth⁡(2x4)4for x2>82atanh⁡(2x4)4for x2<8+constant\begin{cases} \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{4} \right)}}{4} & \text{for}\: x^{2} > 8 \\\frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{4} \right)}}{4} & \text{for}\: x^{2} < 8 \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                   //           /    ___\            \
                   ||  ___      |x*\/ 2 |            |
                   ||\/ 2 *acoth|-------|            |
  /                ||           \   4   /       2    |
 |                 ||--------------------  for x  > 8|
 |   1             ||         4                      |
 | ------ dx = C + |<                                |
 |      2          ||           /    ___\            |
 | 8 - x           ||  ___      |x*\/ 2 |            |
 |                 ||\/ 2 *atanh|-------|            |
/                  ||           \   4   /       2    |
                   ||--------------------  for x  < 8|
                   \\         4                      /
∫18−x2 dx=C+{2acoth⁡(2x4)4for x2>82atanh⁡(2x4)4for x2<8\int \frac{1}{8 - x^{2}}\, dx = C + \begin{cases} \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{4} \right)}}{4} & \text{for}\: x^{2} > 8 \\\frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{4} \right)}}{4} & \text{for}\: x^{2} < 8 \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.120.16
Trazar el gráfico f(x)
Respuesta [src] 
    ___ /          /         ___\\     ___    /    ___\     ___ /          /    ___\\     ___    /        ___\
  \/ 2 *\pi*I + log\-1 + 2*\/ 2 //   \/ 2 *log\2*\/ 2 /   \/ 2 *\pi*I + log\2*\/ 2 //   \/ 2 *log\1 + 2*\/ 2 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
                 8                           8                         8                          8
−2log⁡(22)8+2log⁡(1+22)8−2(log⁡(−1+22)+iπ)8+2(log⁡(22)+iπ)8- \frac{\sqrt{2} \log{\left(2 \sqrt{2} \right)}}{8} + \frac{\sqrt{2} \log{\left(1 + 2 \sqrt{2} \right)}}{8} - \frac{\sqrt{2} \left(\log{\left(-1 + 2 \sqrt{2} \right)} + i \pi\right)}{8} + \frac{\sqrt{2} \left(\log{\left(2 \sqrt{2} \right)} + i \pi\right)}{8} 
=
= 
    ___ /          /         ___\\     ___    /    ___\     ___ /          /    ___\\     ___    /        ___\
  \/ 2 *\pi*I + log\-1 + 2*\/ 2 //   \/ 2 *log\2*\/ 2 /   \/ 2 *\pi*I + log\2*\/ 2 //   \/ 2 *log\1 + 2*\/ 2 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
                 8                           8                         8                          8
−2log⁡(22)8+2log⁡(1+22)8−2(log⁡(−1+22)+iπ)8+2(log⁡(22)+iπ)8- \frac{\sqrt{2} \log{\left(2 \sqrt{2} \right)}}{8} + \frac{\sqrt{2} \log{\left(1 + 2 \sqrt{2} \right)}}{8} - \frac{\sqrt{2} \left(\log{\left(-1 + 2 \sqrt{2} \right)} + i \pi\right)}{8} + \frac{\sqrt{2} \left(\log{\left(2 \sqrt{2} \right)} + i \pi\right)}{8} 
-sqrt(2)*(pi*i + log(-1 + 2*sqrt(2)))/8 - sqrt(2)*log(2*sqrt(2))/8 + sqrt(2)*(pi*i + log(2*sqrt(2)))/8 + sqrt(2)*log(1 + 2*sqrt(2))/8
Respuesta numérica [src] 
0.13063761434512
0.13063761434512

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

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