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Resolución de integrales/ x^2-9/ 1/(8*x^2-9)

Integral de 1/(8*x^2-9) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |     1       
 |  -------- dx
 |     2       
 |  8*x  - 9   
 |             
/              
0
$$\int\limits_{0}^{1} \frac{1}{8 x^{2} - 9}\, dx$$ 
Integral(1/(8*x^2 - 9), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=8, c=-9, context=1/(8*x**2 - 9), symbol=x), False), (ArccothRule(a=1, b=8, c=-9, context=1/(8*x**2 - 9), symbol=x), x**2 > 9/8), (ArctanhRule(a=1, b=8, c=-9, context=1/(8*x**2 - 9), symbol=x), x**2 < 9/8)], context=1/(8*x**2 - 9), symbol=x)

  1. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
                     //            /      ___\               \
                     ||   ___      |2*x*\/ 2 |               |
                     ||-\/ 2 *acoth|---------|               |
  /                  ||            \    3    /        2      |
 |                   ||------------------------  for x  > 9/8|
 |    1              ||           12                         |
 | -------- dx = C + |<                                      |
 |    2              ||            /      ___\               |
 | 8*x  - 9          ||   ___      |2*x*\/ 2 |               |
 |                   ||-\/ 2 *atanh|---------|               |
/                    ||            \    3    /        2      |
                     ||------------------------  for x  < 9/8|
                     \\           12                         /
$$\int \frac{1}{8 x^{2} - 9}\, dx = C + \begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{2 \sqrt{2} x}{3} \right)}}{12} & \text{for}\: x^{2} > \frac{9}{8} \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{2 \sqrt{2} x}{3} \right)}}{12} & \text{for}\: x^{2} < \frac{9}{8} \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
        /          /    ___\\            /        ___\         /          /         ___\\            /    ___\
    ___ |          |3*\/ 2 ||     ___    |    3*\/ 2 |     ___ |          |     3*\/ 2 ||     ___    |3*\/ 2 |
  \/ 2 *|pi*I + log|-------||   \/ 2 *log|1 + -------|   \/ 2 *|pi*I + log|-1 + -------||   \/ 2 *log|-------|
        \          \   4   //            \       4   /         \          \        4   //            \   4   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               24                         24                            24                          24
$$- \frac{\sqrt{2} \log{\left(1 + \frac{3 \sqrt{2}}{4} \right)}}{24} + \frac{\sqrt{2} \log{\left(\frac{3 \sqrt{2}}{4} \right)}}{24} - \frac{\sqrt{2} \left(\log{\left(\frac{3 \sqrt{2}}{4} \right)} + i \pi\right)}{24} + \frac{\sqrt{2} \left(\log{\left(-1 + \frac{3 \sqrt{2}}{4} \right)} + i \pi\right)}{24}$$ 
=
= 
        /          /    ___\\            /        ___\         /          /         ___\\            /    ___\
    ___ |          |3*\/ 2 ||     ___    |    3*\/ 2 |     ___ |          |     3*\/ 2 ||     ___    |3*\/ 2 |
  \/ 2 *|pi*I + log|-------||   \/ 2 *log|1 + -------|   \/ 2 *|pi*I + log|-1 + -------||   \/ 2 *log|-------|
        \          \   4   //            \       4   /         \          \        4   //            \   4   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               24                         24                            24                          24
$$- \frac{\sqrt{2} \log{\left(1 + \frac{3 \sqrt{2}}{4} \right)}}{24} + \frac{\sqrt{2} \log{\left(\frac{3 \sqrt{2}}{4} \right)}}{24} - \frac{\sqrt{2} \left(\log{\left(\frac{3 \sqrt{2}}{4} \right)} + i \pi\right)}{24} + \frac{\sqrt{2} \left(\log{\left(-1 + \frac{3 \sqrt{2}}{4} \right)} + i \pi\right)}{24}$$ 
-sqrt(2)*(pi*i + log(3*sqrt(2)/4))/24 - sqrt(2)*log(1 + 3*sqrt(2)/4)/24 + sqrt(2)*(pi*i + log(-1 + 3*sqrt(2)/4))/24 + sqrt(2)*log(3*sqrt(2)/4)/24
Respuesta numérica [src] 
-0.20774174671341
-0.20774174671341

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

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