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Resolución de integrales/ x^2-6/ dx/(5*x^2-6)

Integral de dx/(5*x^2-6) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |     1       
 |  -------- dx
 |     2       
 |  5*x  - 6   
 |             
/              
0
∫0115x2−6 dx\int\limits_{0}^{1} \frac{1}{5 x^{2} - 6}\, dx 
Integral(1/(5*x^2 - 6), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=5, c=-6, context=1/(5*x**2 - 6), symbol=x), False), (ArccothRule(a=1, b=5, c=-6, context=1/(5*x**2 - 6), symbol=x), x**2 > 6/5), (ArctanhRule(a=1, b=5, c=-6, context=1/(5*x**2 - 6), symbol=x), x**2 < 6/5)], context=1/(5*x**2 - 6), symbol=x)

  1. Añadimos la constante de integración:

    {−30acoth⁡(30x6)30for x2>65−30atanh⁡(30x6)30for x2<65+constant\begin{cases} - \frac{\sqrt{30} \operatorname{acoth}{\left(\frac{\sqrt{30} x}{6} \right)}}{30} & \text{for}\: x^{2} > \frac{6}{5} \\- \frac{\sqrt{30} \operatorname{atanh}{\left(\frac{\sqrt{30} x}{6} \right)}}{30} & \text{for}\: x^{2} < \frac{6}{5} \end{cases}+ \mathrm{constant}

Respuesta:

{−30acoth⁡(30x6)30for x2>65−30atanh⁡(30x6)30for x2<65+constant\begin{cases} - \frac{\sqrt{30} \operatorname{acoth}{\left(\frac{\sqrt{30} x}{6} \right)}}{30} & \text{for}\: x^{2} > \frac{6}{5} \\- \frac{\sqrt{30} \operatorname{atanh}{\left(\frac{\sqrt{30} x}{6} \right)}}{30} & \text{for}\: x^{2} < \frac{6}{5} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                     //             /    ____\               \
                     ||   ____      |x*\/ 30 |               |
                     ||-\/ 30 *acoth|--------|               |
  /                  ||             \   6    /        2      |
 |                   ||------------------------  for x  > 6/5|
 |    1              ||           30                         |
 | -------- dx = C + |<                                      |
 |    2              ||             /    ____\               |
 | 5*x  - 6          ||   ____      |x*\/ 30 |               |
 |                   ||-\/ 30 *atanh|--------|               |
/                    ||             \   6    /        2      |
                     ||------------------------  for x  < 6/5|
                     \\           30                         /
∫15x2−6 dx=C+{−30acoth⁡(30x6)30for x2>65−30atanh⁡(30x6)30for x2<65\int \frac{1}{5 x^{2} - 6}\, dx = C + \begin{cases} - \frac{\sqrt{30} \operatorname{acoth}{\left(\frac{\sqrt{30} x}{6} \right)}}{30} & \text{for}\: x^{2} > \frac{6}{5} \\- \frac{\sqrt{30} \operatorname{atanh}{\left(\frac{\sqrt{30} x}{6} \right)}}{30} & \text{for}\: x^{2} < \frac{6}{5} \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] 
         /          /  ____\\             /      ____\          /          /       ____\\             /  ____\
    ____ |          |\/ 30 ||     ____    |    \/ 30 |     ____ |          |     \/ 30 ||     ____    |\/ 30 |
  \/ 30 *|pi*I + log|------||   \/ 30 *log|1 + ------|   \/ 30 *|pi*I + log|-1 + ------||   \/ 30 *log|------|
         \          \  5   //             \      5   /          \          \       5   //             \  5   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               60                         60                            60                          60
−30log⁡(1+305)60+30log⁡(305)60−30(log⁡(305)+iπ)60+30(log⁡(−1+305)+iπ)60- \frac{\sqrt{30} \log{\left(1 + \frac{\sqrt{30}}{5} \right)}}{60} + \frac{\sqrt{30} \log{\left(\frac{\sqrt{30}}{5} \right)}}{60} - \frac{\sqrt{30} \left(\log{\left(\frac{\sqrt{30}}{5} \right)} + i \pi\right)}{60} + \frac{\sqrt{30} \left(\log{\left(-1 + \frac{\sqrt{30}}{5} \right)} + i \pi\right)}{60} 
=
= 
         /          /  ____\\             /      ____\          /          /       ____\\             /  ____\
    ____ |          |\/ 30 ||     ____    |    \/ 30 |     ____ |          |     \/ 30 ||     ____    |\/ 30 |
  \/ 30 *|pi*I + log|------||   \/ 30 *log|1 + ------|   \/ 30 *|pi*I + log|-1 + ------||   \/ 30 *log|------|
         \          \  5   //             \      5   /          \          \       5   //             \  5   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               60                         60                            60                          60
−30log⁡(1+305)60+30log⁡(305)60−30(log⁡(305)+iπ)60+30(log⁡(−1+305)+iπ)60- \frac{\sqrt{30} \log{\left(1 + \frac{\sqrt{30}}{5} \right)}}{60} + \frac{\sqrt{30} \log{\left(\frac{\sqrt{30}}{5} \right)}}{60} - \frac{\sqrt{30} \left(\log{\left(\frac{\sqrt{30}}{5} \right)} + i \pi\right)}{60} + \frac{\sqrt{30} \left(\log{\left(-1 + \frac{\sqrt{30}}{5} \right)} + i \pi\right)}{60} 
-sqrt(30)*(pi*i + log(sqrt(30)/5))/60 - sqrt(30)*log(1 + sqrt(30)/5)/60 + sqrt(30)*(pi*i + log(-1 + sqrt(30)/5))/60 + sqrt(30)*log(sqrt(30)/5)/60
Respuesta numérica [src] 
-0.281983082722996
-0.281983082722996

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

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