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Resolución de integrales/ 1/(7x^2-8)

Integral de 1/(7x^2-8) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |     1       
 |  -------- dx
 |     2       
 |  7*x  - 8   
 |             
/              
0
∫0117x2−8 dx\int\limits_{0}^{1} \frac{1}{7 x^{2} - 8}\, dx 
Integral(1/(7*x^2 - 8), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=7, c=-8, context=1/(7*x**2 - 8), symbol=x), False), (ArccothRule(a=1, b=7, c=-8, context=1/(7*x**2 - 8), symbol=x), x**2 > 8/7), (ArctanhRule(a=1, b=7, c=-8, context=1/(7*x**2 - 8), symbol=x), x**2 < 8/7)], context=1/(7*x**2 - 8), symbol=x)

  1. Añadimos la constante de integración:

    {−14acoth⁡(14x4)28for x2>87−14atanh⁡(14x4)28for x2<87+constant\begin{cases} - \frac{\sqrt{14} \operatorname{acoth}{\left(\frac{\sqrt{14} x}{4} \right)}}{28} & \text{for}\: x^{2} > \frac{8}{7} \\- \frac{\sqrt{14} \operatorname{atanh}{\left(\frac{\sqrt{14} x}{4} \right)}}{28} & \text{for}\: x^{2} < \frac{8}{7} \end{cases}+ \mathrm{constant}

Respuesta:

{−14acoth⁡(14x4)28for x2>87−14atanh⁡(14x4)28for x2<87+constant\begin{cases} - \frac{\sqrt{14} \operatorname{acoth}{\left(\frac{\sqrt{14} x}{4} \right)}}{28} & \text{for}\: x^{2} > \frac{8}{7} \\- \frac{\sqrt{14} \operatorname{atanh}{\left(\frac{\sqrt{14} x}{4} \right)}}{28} & \text{for}\: x^{2} < \frac{8}{7} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                     //             /    ____\               \
                     ||   ____      |x*\/ 14 |               |
                     ||-\/ 14 *acoth|--------|               |
  /                  ||             \   4    /        2      |
 |                   ||------------------------  for x  > 8/7|
 |    1              ||           28                         |
 | -------- dx = C + |<                                      |
 |    2              ||             /    ____\               |
 | 7*x  - 8          ||   ____      |x*\/ 14 |               |
 |                   ||-\/ 14 *atanh|--------|               |
/                    ||             \   4    /        2      |
                     ||------------------------  for x  < 8/7|
                     \\           28                         /
∫17x2−8 dx=C+{−14acoth⁡(14x4)28for x2>87−14atanh⁡(14x4)28for x2<87\int \frac{1}{7 x^{2} - 8}\, dx = C + \begin{cases} - \frac{\sqrt{14} \operatorname{acoth}{\left(\frac{\sqrt{14} x}{4} \right)}}{28} & \text{for}\: x^{2} > \frac{8}{7} \\- \frac{\sqrt{14} \operatorname{atanh}{\left(\frac{\sqrt{14} x}{4} \right)}}{28} & \text{for}\: x^{2} < \frac{8}{7} \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-2.00.0
Trazar el gráfico f(x)
Respuesta [src] 
         /          /    ____\\             /        ____\          /          /         ____\\             /    ____\
    ____ |          |2*\/ 14 ||     ____    |    2*\/ 14 |     ____ |          |     2*\/ 14 ||     ____    |2*\/ 14 |
  \/ 14 *|pi*I + log|--------||   \/ 14 *log|1 + --------|   \/ 14 *|pi*I + log|-1 + --------||   \/ 14 *log|--------|
         \          \   7    //             \       7    /          \          \        7    //             \   7    /
- ----------------------------- - ------------------------ + ---------------------------------- + --------------------
                56                           56                              56                            56
−14log⁡(1+2147)56+14log⁡(2147)56−14(log⁡(2147)+iπ)56+14(log⁡(−1+2147)+iπ)56- \frac{\sqrt{14} \log{\left(1 + \frac{2 \sqrt{14}}{7} \right)}}{56} + \frac{\sqrt{14} \log{\left(\frac{2 \sqrt{14}}{7} \right)}}{56} - \frac{\sqrt{14} \left(\log{\left(\frac{2 \sqrt{14}}{7} \right)} + i \pi\right)}{56} + \frac{\sqrt{14} \left(\log{\left(-1 + \frac{2 \sqrt{14}}{7} \right)} + i \pi\right)}{56} 
=
= 
         /          /    ____\\             /        ____\          /          /         ____\\             /    ____\
    ____ |          |2*\/ 14 ||     ____    |    2*\/ 14 |     ____ |          |     2*\/ 14 ||     ____    |2*\/ 14 |
  \/ 14 *|pi*I + log|--------||   \/ 14 *log|1 + --------|   \/ 14 *|pi*I + log|-1 + --------||   \/ 14 *log|--------|
         \          \   7    //             \       7    /          \          \        7    //             \   7    /
- ----------------------------- - ------------------------ + ---------------------------------- + --------------------
                56                           56                              56                            56
−14log⁡(1+2147)56+14log⁡(2147)56−14(log⁡(2147)+iπ)56+14(log⁡(−1+2147)+iπ)56- \frac{\sqrt{14} \log{\left(1 + \frac{2 \sqrt{14}}{7} \right)}}{56} + \frac{\sqrt{14} \log{\left(\frac{2 \sqrt{14}}{7} \right)}}{56} - \frac{\sqrt{14} \left(\log{\left(\frac{2 \sqrt{14}}{7} \right)} + i \pi\right)}{56} + \frac{\sqrt{14} \left(\log{\left(-1 + \frac{2 \sqrt{14}}{7} \right)} + i \pi\right)}{56} 
-sqrt(14)*(pi*i + log(2*sqrt(14)/7))/56 - sqrt(14)*log(1 + 2*sqrt(14)/7)/56 + sqrt(14)*(pi*i + log(-1 + 2*sqrt(14)/7))/56 + sqrt(14)*log(2*sqrt(14)/7)/56
Respuesta numérica [src] 
-0.227177695780752
-0.227177695780752

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

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