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Resolución de integrales/ 2*x^2/ dx/(3-2*x^2)

Integral de dx/(3-2*x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

  1. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
                     //           /    ___\              \
                     ||  ___      |x*\/ 6 |              |
                     ||\/ 6 *acoth|-------|              |
  /                  ||           \   3   /       2      |
 |                   ||--------------------  for x  > 3/2|
 |    1              ||         6                        |
 | -------- dx = C + |<                                  |
 |        2          ||           /    ___\              |
 | 3 - 2*x           ||  ___      |x*\/ 6 |              |
 |                   ||\/ 6 *atanh|-------|              |
/                    ||           \   3   /       2      |
                     ||--------------------  for x  < 3/2|
                     \\         6                        /
$$\int \frac{1}{3 - 2 x^{2}}\, dx = C + \begin{cases} \frac{\sqrt{6} \operatorname{acoth}{\left(\frac{\sqrt{6} x}{3} \right)}}{6} & \text{for}\: x^{2} > \frac{3}{2} \\\frac{\sqrt{6} \operatorname{atanh}{\left(\frac{\sqrt{6} x}{3} \right)}}{6} & \text{for}\: x^{2} < \frac{3}{2} \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
        /          /       ___\\            /  ___\         /          /  ___\\            /      ___\
    ___ |          |     \/ 6 ||     ___    |\/ 6 |     ___ |          |\/ 6 ||     ___    |    \/ 6 |
  \/ 6 *|pi*I + log|-1 + -----||   \/ 6 *log|-----|   \/ 6 *|pi*I + log|-----||   \/ 6 *log|1 + -----|
        \          \       2  //            \  2  /         \          \  2  //            \      2  /
- ------------------------------ - ---------------- + ------------------------- + --------------------
                12                        12                      12                       12
$$- \frac{\sqrt{6} \log{\left(\frac{\sqrt{6}}{2} \right)}}{12} + \frac{\sqrt{6} \log{\left(1 + \frac{\sqrt{6}}{2} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(-1 + \frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(\frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12}$$ 
=
= 
        /          /       ___\\            /  ___\         /          /  ___\\            /      ___\
    ___ |          |     \/ 6 ||     ___    |\/ 6 |     ___ |          |\/ 6 ||     ___    |    \/ 6 |
  \/ 6 *|pi*I + log|-1 + -----||   \/ 6 *log|-----|   \/ 6 *|pi*I + log|-----||   \/ 6 *log|1 + -----|
        \          \       2  //            \  2  /         \          \  2  //            \      2  /
- ------------------------------ - ---------------- + ------------------------- + --------------------
                12                        12                      12                       12
$$- \frac{\sqrt{6} \log{\left(\frac{\sqrt{6}}{2} \right)}}{12} + \frac{\sqrt{6} \log{\left(1 + \frac{\sqrt{6}}{2} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(-1 + \frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(\frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12}$$ 
-sqrt(6)*(pi*i + log(-1 + sqrt(6)/2))/12 - sqrt(6)*log(sqrt(6)/2)/12 + sqrt(6)*(pi*i + log(sqrt(6)/2))/12 + sqrt(6)*log(1 + sqrt(6)/2)/12
Respuesta numérica [src] 
0.467940655051785
0.467940655051785

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

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