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Resolución de integrales/ -5x^2/ 1/((-5x^2)+13)

Integral de 1/((-5x^2)+13) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |       1        
 |  ----------- dx
 |       2        
 |  - 5*x  + 13   
 |                
/                 
0
$$\int\limits_{0}^{1} \frac{1}{13 - 5 x^{2}}\, dx$$ 
Integral(1/(-5*x^2 + 13), (x, 0, 1))
Solución detallada

    PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-5, c=13, context=1/(13 - 5*x**2), symbol=x), False), (ArccothRule(a=1, b=-5, c=13, context=1/(13 - 5*x**2), symbol=x), x**2 > 13/5), (ArctanhRule(a=1, b=-5, c=13, context=1/(13 - 5*x**2), symbol=x), x**2 < 13/5)], context=1/(13 - 5*x**2), symbol=x)

  1. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
                        //            /    ____\               \
                        ||  ____      |x*\/ 65 |               |
                        ||\/ 65 *acoth|--------|               |
  /                     ||            \   13   /       2       |
 |                      ||----------------------  for x  > 13/5|
 |      1               ||          65                         |
 | ----------- dx = C + |<                                     |
 |      2               ||            /    ____\               |
 | - 5*x  + 13          ||  ____      |x*\/ 65 |               |
 |                      ||\/ 65 *atanh|--------|               |
/                       ||            \   13   /       2       |
                        ||----------------------  for x  < 13/5|
                        \\          65                         /
$$\int \frac{1}{13 - 5 x^{2}}\, dx = C + \begin{cases} \frac{\sqrt{65} \operatorname{acoth}{\left(\frac{\sqrt{65} x}{13} \right)}}{65} & \text{for}\: x^{2} > \frac{13}{5} \\\frac{\sqrt{65} \operatorname{atanh}{\left(\frac{\sqrt{65} x}{13} \right)}}{65} & \text{for}\: x^{2} < \frac{13}{5} \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
         /          /       ____\\             /  ____\          /          /  ____\\             /      ____\
    ____ |          |     \/ 65 ||     ____    |\/ 65 |     ____ |          |\/ 65 ||     ____    |    \/ 65 |
  \/ 65 *|pi*I + log|-1 + ------||   \/ 65 *log|------|   \/ 65 *|pi*I + log|------||   \/ 65 *log|1 + ------|
         \          \       5   //             \  5   /          \          \  5   //             \      5   /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
                130                         130                       130                        130
$$- \frac{\sqrt{65} \log{\left(\frac{\sqrt{65}}{5} \right)}}{130} + \frac{\sqrt{65} \log{\left(1 + \frac{\sqrt{65}}{5} \right)}}{130} - \frac{\sqrt{65} \left(\log{\left(-1 + \frac{\sqrt{65}}{5} \right)} + i \pi\right)}{130} + \frac{\sqrt{65} \left(\log{\left(\frac{\sqrt{65}}{5} \right)} + i \pi\right)}{130}$$ 
=
= 
         /          /       ____\\             /  ____\          /          /  ____\\             /      ____\
    ____ |          |     \/ 65 ||     ____    |\/ 65 |     ____ |          |\/ 65 ||     ____    |    \/ 65 |
  \/ 65 *|pi*I + log|-1 + ------||   \/ 65 *log|------|   \/ 65 *|pi*I + log|------||   \/ 65 *log|1 + ------|
         \          \       5   //             \  5   /          \          \  5   //             \      5   /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
                130                         130                       130                        130
$$- \frac{\sqrt{65} \log{\left(\frac{\sqrt{65}}{5} \right)}}{130} + \frac{\sqrt{65} \log{\left(1 + \frac{\sqrt{65}}{5} \right)}}{130} - \frac{\sqrt{65} \left(\log{\left(-1 + \frac{\sqrt{65}}{5} \right)} + i \pi\right)}{130} + \frac{\sqrt{65} \left(\log{\left(\frac{\sqrt{65}}{5} \right)} + i \pi\right)}{130}$$ 
-sqrt(65)*(pi*i + log(-1 + sqrt(65)/5))/130 - sqrt(65)*log(sqrt(65)/5)/130 + sqrt(65)*(pi*i + log(sqrt(65)/5))/130 + sqrt(65)*log(1 + sqrt(65)/5)/130
Respuesta numérica [src] 
0.0899608124119269
0.0899608124119269

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

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