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Resolución de integrales/ (3x^2)/ 1/((3x^2)-5)

Integral de 1/((3x^2)-5) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |     1       
 |  -------- dx
 |     2       
 |  3*x  - 5   
 |             
/              
0
∫0113x2−5 dx\int\limits_{0}^{1} \frac{1}{3 x^{2} - 5}\, dx 
Integral(1/(3*x^2 - 5), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=3, c=-5, context=1/(3*x**2 - 5), symbol=x), False), (ArccothRule(a=1, b=3, c=-5, context=1/(3*x**2 - 5), symbol=x), x**2 > 5/3), (ArctanhRule(a=1, b=3, c=-5, context=1/(3*x**2 - 5), symbol=x), x**2 < 5/3)], context=1/(3*x**2 - 5), symbol=x)

  1. Añadimos la constante de integración:

    {−15acoth⁡(15x5)15for x2>53−15atanh⁡(15x5)15for x2<53+constant\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left(\frac{\sqrt{15} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left(\frac{\sqrt{15} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{5}{3} \end{cases}+ \mathrm{constant}

Respuesta:

{−15acoth⁡(15x5)15for x2>53−15atanh⁡(15x5)15for x2<53+constant\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left(\frac{\sqrt{15} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left(\frac{\sqrt{15} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{5}{3} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                     //             /    ____\               \
                     ||   ____      |x*\/ 15 |               |
                     ||-\/ 15 *acoth|--------|               |
  /                  ||             \   5    /        2      |
 |                   ||------------------------  for x  > 5/3|
 |    1              ||           15                         |
 | -------- dx = C + |<                                      |
 |    2              ||             /    ____\               |
 | 3*x  - 5          ||   ____      |x*\/ 15 |               |
 |                   ||-\/ 15 *atanh|--------|               |
/                    ||             \   5    /        2      |
                     ||------------------------  for x  < 5/3|
                     \\           15                         /
∫13x2−5 dx=C+{−15acoth⁡(15x5)15for x2>53−15atanh⁡(15x5)15for x2<53\int \frac{1}{3 x^{2} - 5}\, dx = C + \begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left(\frac{\sqrt{15} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left(\frac{\sqrt{15} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{5}{3} \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-1.00.0
Trazar el gráfico f(x)
Respuesta [src] 
         /          /  ____\\             /      ____\          /          /       ____\\             /  ____\
    ____ |          |\/ 15 ||     ____    |    \/ 15 |     ____ |          |     \/ 15 ||     ____    |\/ 15 |
  \/ 15 *|pi*I + log|------||   \/ 15 *log|1 + ------|   \/ 15 *|pi*I + log|-1 + ------||   \/ 15 *log|------|
         \          \  3   //             \      3   /          \          \       3   //             \  3   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               30                         30                            30                          30
−15log⁡(1+153)30+15log⁡(153)30−15(log⁡(153)+iπ)30+15(log⁡(−1+153)+iπ)30- \frac{\sqrt{15} \log{\left(1 + \frac{\sqrt{15}}{3} \right)}}{30} + \frac{\sqrt{15} \log{\left(\frac{\sqrt{15}}{3} \right)}}{30} - \frac{\sqrt{15} \left(\log{\left(\frac{\sqrt{15}}{3} \right)} + i \pi\right)}{30} + \frac{\sqrt{15} \left(\log{\left(-1 + \frac{\sqrt{15}}{3} \right)} + i \pi\right)}{30} 
=
= 
         /          /  ____\\             /      ____\          /          /       ____\\             /  ____\
    ____ |          |\/ 15 ||     ____    |    \/ 15 |     ____ |          |     \/ 15 ||     ____    |\/ 15 |
  \/ 15 *|pi*I + log|------||   \/ 15 *log|1 + ------|   \/ 15 *|pi*I + log|-1 + ------||   \/ 15 *log|------|
         \          \  3   //             \      3   /          \          \       3   //             \  3   /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
               30                         30                            30                          30
−15log⁡(1+153)30+15log⁡(153)30−15(log⁡(153)+iπ)30+15(log⁡(−1+153)+iπ)30- \frac{\sqrt{15} \log{\left(1 + \frac{\sqrt{15}}{3} \right)}}{30} + \frac{\sqrt{15} \log{\left(\frac{\sqrt{15}}{3} \right)}}{30} - \frac{\sqrt{15} \left(\log{\left(\frac{\sqrt{15}}{3} \right)} + i \pi\right)}{30} + \frac{\sqrt{15} \left(\log{\left(-1 + \frac{\sqrt{15}}{3} \right)} + i \pi\right)}{30} 
-sqrt(15)*(pi*i + log(sqrt(15)/3))/30 - sqrt(15)*log(1 + sqrt(15)/3)/30 + sqrt(15)*(pi*i + log(-1 + sqrt(15)/3))/30 + sqrt(15)*log(sqrt(15)/3)/30
Respuesta numérica [src] 
-0.266388580125985
-0.266388580125985

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

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