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Resolución de integrales/ 2-x^2/ x^2/(2-x^2)^2

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1             
  /             
 |              
 |       2      
 |      x       
 |  --------- dx
 |          2   
 |  /     2\    
 |  \2 - x /    
 |              
/               
0
∫01x2(2−x2)2 dx\int\limits_{0}^{1} \frac{x^{2}}{\left(2 - x^{2}\right)^{2}}\, dx 
Integral(x^2/(2 - x^2)^2, (x, 0, 1))
Respuesta (Indefinida) [src] 
                                                                                                                                                 //            /    ___\             \
                        //      /                                          /         ___\      /        ___\\                                \   ||   ___      |x*\/ 2 |             |
  /                     ||      |                                          |     x*\/ 2 |      |    x*\/ 2 ||                                |   ||-\/ 2 *acoth|-------|             |
 |                      ||      |                                       log|-1 + -------|   log|1 + -------||                                |   ||            \   2   /        2    |
 |      2               ||  ___ |         1                 1              \        2   /      \       2   /|                                |   ||----------------------  for x  > 2|
 |     x                ||\/ 2 *|- --------------- - ---------------- - ----------------- + ----------------|                                |   ||          2                       |
 | --------- dx = C + 2*|<      |    /        ___\     /         ___\           4                  4        |                                | + |<                                  |
 |         2            ||      |    |    x*\/ 2 |     |     x*\/ 2 |                                       |                                |   ||            /    ___\             |
 | /     2\             ||      |  4*|1 + -------|   4*|-1 + -------|                                       |                                |   ||   ___      |x*\/ 2 |             |
 | \2 - x /             ||      \    \       2   /     \        2   /                                       /         /       ___        ___\|   ||-\/ 2 *atanh|-------|             |
 |                      ||-----------------------------------------------------------------------------------  for And\x > -\/ 2 , x < \/ 2 /|   ||            \   2   /        2    |
/                       \\                                         4                                                                         /   ||----------------------  for x  < 2|
                                                                                                                                                 \\          2                       /
∫x2(2−x2)2 dx=C+2({2(−log⁡(2x2−1)4+log⁡(2x2+1)4−14(2x2+1)−14(2x2−1))4for x>−2∧x<2)+{−2acoth⁡(2x2)2for x2>2−2atanh⁡(2x2)2for x2<2\int \frac{x^{2}}{\left(2 - x^{2}\right)^{2}}\, dx = C + 2 \left(\begin{cases} \frac{\sqrt{2} \left(- \frac{\log{\left(\frac{\sqrt{2} x}{2} - 1 \right)}}{4} + \frac{\log{\left(\frac{\sqrt{2} x}{2} + 1 \right)}}{4} - \frac{1}{4 \left(\frac{\sqrt{2} x}{2} + 1\right)} - \frac{1}{4 \left(\frac{\sqrt{2} x}{2} - 1\right)}\right)}{4} & \text{for}\: x > - \sqrt{2} \wedge x < \sqrt{2} \end{cases}\right) + \begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} < 2 \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.9002
Trazar el gráfico f(x)
Respuesta [src] 
      ___ /          /  ___\\     ___    /      ___\     ___ /          /       ___\\     ___    /  ___\
1   \/ 2 *\pi*I + log\\/ 2 //   \/ 2 *log\1 + \/ 2 /   \/ 2 *\pi*I + log\-1 + \/ 2 //   \/ 2 *log\\/ 2 /
- - ------------------------- - -------------------- + ------------------------------ + ----------------
2               8                        8                           8                         8
−2log⁡(1+2)8+2log⁡(2)8+12−2(log⁡(2)+iπ)8+2(log⁡(−1+2)+iπ)8- \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{8} + \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{8} + \frac{1}{2} - \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{8} + \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{8} 
=
= 
      ___ /          /  ___\\     ___    /      ___\     ___ /          /       ___\\     ___    /  ___\
1   \/ 2 *\pi*I + log\\/ 2 //   \/ 2 *log\1 + \/ 2 /   \/ 2 *\pi*I + log\-1 + \/ 2 //   \/ 2 *log\\/ 2 /
- - ------------------------- - -------------------- + ------------------------------ + ----------------
2               8                        8                           8                         8
−2log⁡(1+2)8+2log⁡(2)8+12−2(log⁡(2)+iπ)8+2(log⁡(−1+2)+iπ)8- \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{8} + \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{8} + \frac{1}{2} - \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{8} + \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{8} 
1/2 - sqrt(2)*(pi*i + log(sqrt(2)))/8 - sqrt(2)*log(1 + sqrt(2))/8 + sqrt(2)*(pi*i + log(-1 + sqrt(2)))/8 + sqrt(2)*log(sqrt(2))/8
Respuesta numérica [src] 
0.188387379929885
0.188387379929885

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

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