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Resolución de integrales/ 2*x^2/ sqrt(14)/(2*x^2-7)

Integral de sqrt(14)/(2*x^2-7) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1            
  /            
 |             
 |     ____    
 |   \/ 14     
 |  -------- dx
 |     2       
 |  2*x  - 7   
 |             
/              
0
∫01142x2−7 dx\int\limits_{0}^{1} \frac{\sqrt{14}}{2 x^{2} - 7}\, dx 
Integral(sqrt(14)/(2*x^2 - 7), (x, 0, 1))
Solución detallada

  1. La integral del producto de una función por una constante es la constante por la integral de esta función:

    ∫142x2−7 dx=14∫12x2−7 dx\int \frac{\sqrt{14}}{2 x^{2} - 7}\, dx = \sqrt{14} \int \frac{1}{2 x^{2} - 7}\, dx

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

    Por lo tanto, el resultado es: 14({−14acoth⁡(14x7)14for x2>72−14atanh⁡(14x7)14for x2<72)\sqrt{14} \left(\begin{cases} - \frac{\sqrt{14} \operatorname{acoth}{\left(\frac{\sqrt{14} x}{7} \right)}}{14} & \text{for}\: x^{2} > \frac{7}{2} \\- \frac{\sqrt{14} \operatorname{atanh}{\left(\frac{\sqrt{14} x}{7} \right)}}{14} & \text{for}\: x^{2} < \frac{7}{2} \end{cases}\right)

  2. Ahora simplificar:

    {−acoth⁡(14x7)for x2>72−atanh⁡(14x7)for x2<72\begin{cases} - \operatorname{acoth}{\left(\frac{\sqrt{14} x}{7} \right)} & \text{for}\: x^{2} > \frac{7}{2} \\- \operatorname{atanh}{\left(\frac{\sqrt{14} x}{7} \right)} & \text{for}\: x^{2} < \frac{7}{2} \end{cases}

  3. Añadimos la constante de integración:

    {−acoth⁡(14x7)for x2>72−atanh⁡(14x7)for x2<72+constant\begin{cases} - \operatorname{acoth}{\left(\frac{\sqrt{14} x}{7} \right)} & \text{for}\: x^{2} > \frac{7}{2} \\- \operatorname{atanh}{\left(\frac{\sqrt{14} x}{7} \right)} & \text{for}\: x^{2} < \frac{7}{2} \end{cases}+ \mathrm{constant}

Respuesta:

{−acoth⁡(14x7)for x2>72−atanh⁡(14x7)for x2<72+constant\begin{cases} - \operatorname{acoth}{\left(\frac{\sqrt{14} x}{7} \right)} & \text{for}\: x^{2} > \frac{7}{2} \\- \operatorname{atanh}{\left(\frac{\sqrt{14} x}{7} \right)} & \text{for}\: x^{2} < \frac{7}{2} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                            //             /    ____\               \
                            ||   ____      |x*\/ 14 |               |
  /                         ||-\/ 14 *acoth|--------|               |
 |                          ||             \   7    /        2      |
 |    ____                  ||------------------------  for x  > 7/2|
 |  \/ 14              ____ ||           14                         |
 | -------- dx = C + \/ 14 *|<                                      |
 |    2                     ||             /    ____\               |
 | 2*x  - 7                 ||   ____      |x*\/ 14 |               |
 |                          ||-\/ 14 *atanh|--------|               |
/                           ||             \   7    /        2      |
                            ||------------------------  for x  < 7/2|
                            \\           14                         /
∫142x2−7 dx=C+14({−14acoth⁡(14x7)14for x2>72−14atanh⁡(14x7)14for x2<72)\int \frac{\sqrt{14}}{2 x^{2} - 7}\, dx = C + \sqrt{14} \left(\begin{cases} - \frac{\sqrt{14} \operatorname{acoth}{\left(\frac{\sqrt{14} x}{7} \right)}}{14} & \text{for}\: x^{2} > \frac{7}{2} \\- \frac{\sqrt{14} \operatorname{atanh}{\left(\frac{\sqrt{14} x}{7} \right)}}{14} & \text{for}\: x^{2} < \frac{7}{2} \end{cases}\right)
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90-1.00-0.50
Trazar el gráfico f(x)
Respuesta [src] 
       /            /      ____\          /          /       ____\\\          /            /  ____\          /          /  ____\\\
       |    ____    |    \/ 14 |     ____ |          |     \/ 14 |||          |    ____    |\/ 14 |     ____ |          |\/ 14 |||
       |  \/ 14 *log|1 + ------|   \/ 14 *|pi*I + log|-1 + ------|||          |  \/ 14 *log|------|   \/ 14 *|pi*I + log|------|||
  ____ |            \      2   /          \          \       2   //|     ____ |            \  2   /          \          \  2   //|
\/ 14 *|- ---------------------- + --------------------------------| - \/ 14 *|- ------------------ + ---------------------------|
       \            28                            28               /          \          28                        28            /
−14(−14log⁡(142)28+14(log⁡(142)+iπ)28)+14(−14log⁡(1+142)28+14(log⁡(−1+142)+iπ)28)- \sqrt{14} \left(- \frac{\sqrt{14} \log{\left(\frac{\sqrt{14}}{2} \right)}}{28} + \frac{\sqrt{14} \left(\log{\left(\frac{\sqrt{14}}{2} \right)} + i \pi\right)}{28}\right) + \sqrt{14} \left(- \frac{\sqrt{14} \log{\left(1 + \frac{\sqrt{14}}{2} \right)}}{28} + \frac{\sqrt{14} \left(\log{\left(-1 + \frac{\sqrt{14}}{2} \right)} + i \pi\right)}{28}\right) 
=
= 
       /            /      ____\          /          /       ____\\\          /            /  ____\          /          /  ____\\\
       |    ____    |    \/ 14 |     ____ |          |     \/ 14 |||          |    ____    |\/ 14 |     ____ |          |\/ 14 |||
       |  \/ 14 *log|1 + ------|   \/ 14 *|pi*I + log|-1 + ------|||          |  \/ 14 *log|------|   \/ 14 *|pi*I + log|------|||
  ____ |            \      2   /          \          \       2   //|     ____ |            \  2   /          \          \  2   //|
\/ 14 *|- ---------------------- + --------------------------------| - \/ 14 *|- ------------------ + ---------------------------|
       \            28                            28               /          \          28                        28            /
−14(−14log⁡(142)28+14(log⁡(142)+iπ)28)+14(−14log⁡(1+142)28+14(log⁡(−1+142)+iπ)28)- \sqrt{14} \left(- \frac{\sqrt{14} \log{\left(\frac{\sqrt{14}}{2} \right)}}{28} + \frac{\sqrt{14} \left(\log{\left(\frac{\sqrt{14}}{2} \right)} + i \pi\right)}{28}\right) + \sqrt{14} \left(- \frac{\sqrt{14} \log{\left(1 + \frac{\sqrt{14}}{2} \right)}}{28} + \frac{\sqrt{14} \left(\log{\left(-1 + \frac{\sqrt{14}}{2} \right)} + i \pi\right)}{28}\right) 
sqrt(14)*(-sqrt(14)*log(1 + sqrt(14)/2)/28 + sqrt(14)*(pi*i + log(-1 + sqrt(14)/2))/28) - sqrt(14)*(-sqrt(14)*log(sqrt(14)/2)/28 + sqrt(14)*(pi*i + log(sqrt(14)/2))/28)
Respuesta numérica [src] 
-0.596455365496524
-0.596455365496524

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

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