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Resolución de integrales/ 1*x^2/ x^2-1/ 1/(81*x^2-16)

Integral de 1/(81*x^2-16) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  4              
  /              
 |               
 |      1        
 |  ---------- dx
 |      2        
 |  81*x  - 16   
 |               
/                
2
∫24181x2−16 dx\int\limits_{2}^{4} \frac{1}{81 x^{2} - 16}\, dx 
Integral(1/(81*x^2 - 16), (x, 2, 4))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=81, c=-16, context=1/(81*x**2 - 16), symbol=x), False), (ArccothRule(a=1, b=81, c=-16, context=1/(81*x**2 - 16), symbol=x), x**2 > 16/81), (ArctanhRule(a=1, b=81, c=-16, context=1/(81*x**2 - 16), symbol=x), x**2 < 16/81)], context=1/(81*x**2 - 16), symbol=x)

  1. Añadimos la constante de integración:

    {−acoth⁡(9x4)36for x2>1681−atanh⁡(9x4)36for x2<1681+constant\begin{cases} - \frac{\operatorname{acoth}{\left(\frac{9 x}{4} \right)}}{36} & \text{for}\: x^{2} > \frac{16}{81} \\- \frac{\operatorname{atanh}{\left(\frac{9 x}{4} \right)}}{36} & \text{for}\: x^{2} < \frac{16}{81} \end{cases}+ \mathrm{constant}

Respuesta:

{−acoth⁡(9x4)36for x2>1681−atanh⁡(9x4)36for x2<1681+constant\begin{cases} - \frac{\operatorname{acoth}{\left(\frac{9 x}{4} \right)}}{36} & \text{for}\: x^{2} > \frac{16}{81} \\- \frac{\operatorname{atanh}{\left(\frac{9 x}{4} \right)}}{36} & \text{for}\: x^{2} < \frac{16}{81} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                       //      /9*x\              \
                       ||-acoth|---|              |
  /                    ||      \ 4 /        2   16|
 |                     ||------------  for x  > --|
 |     1               ||     36                81|
 | ---------- dx = C + |<                         |
 |     2               ||      /9*x\              |
 | 81*x  - 16          ||-atanh|---|              |
 |                     ||      \ 4 /        2   16|
/                      ||------------  for x  < --|
                       \\     36                81/
∫181x2−16 dx=C+{−acoth⁡(9x4)36for x2>1681−atanh⁡(9x4)36for x2<1681\int \frac{1}{81 x^{2} - 16}\, dx = C + \begin{cases} - \frac{\operatorname{acoth}{\left(\frac{9 x}{4} \right)}}{36} & \text{for}\: x^{2} > \frac{16}{81} \\- \frac{\operatorname{atanh}{\left(\frac{9 x}{4} \right)}}{36} & \text{for}\: x^{2} < \frac{16}{81} \end{cases}
Simplificar
Gráfica
2.04.02.22.42.62.83.03.23.43.63.80.01-0.01
Trazar el gráfico f(x)
Respuesta [src] 
  log(14/9)   log(40/9)   log(22/9)   log(32/9)
- --------- - --------- + --------- + ---------
      72          72          72          72
−log⁡(409)72−log⁡(149)72+log⁡(229)72+log⁡(329)72- \frac{\log{\left(\frac{40}{9} \right)}}{72} - \frac{\log{\left(\frac{14}{9} \right)}}{72} + \frac{\log{\left(\frac{22}{9} \right)}}{72} + \frac{\log{\left(\frac{32}{9} \right)}}{72} 
=
= 
  log(14/9)   log(40/9)   log(22/9)   log(32/9)
- --------- - --------- + --------- + ---------
      72          72          72          72
−log⁡(409)72−log⁡(149)72+log⁡(229)72+log⁡(329)72- \frac{\log{\left(\frac{40}{9} \right)}}{72} - \frac{\log{\left(\frac{14}{9} \right)}}{72} + \frac{\log{\left(\frac{22}{9} \right)}}{72} + \frac{\log{\left(\frac{32}{9} \right)}}{72} 
-log(14/9)/72 - log(40/9)/72 + log(22/9)/72 + log(32/9)/72
Respuesta numérica [src] 
0.00317835517262288
0.00317835517262288

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

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