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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1              
  /              
 |               
 |      x        
 |  ---------- dx
 |   2           
 |  x  - x - 3   
 |               
/                
0
∫01x(x2−x)−3 dx\int\limits_{0}^{1} \frac{x}{\left(x^{2} - x\right) - 3}\, dx 
Integral(x/(x^2 - x - 3), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                            //             /    ____           \                         \
                                            ||   ____      |2*\/ 13 *(-1/2 + x)|                         |
                                            ||-\/ 13 *acoth|-------------------|                         |
  /                                         ||             \         13        /                 2       |
 |                        /      2    \     ||-----------------------------------  for (-1/2 + x)  > 13/4|
 |     x               log\-3 + x  - x/     ||                 26                                        |
 | ---------- dx = C + ---------------- + 2*|<                                                           |
 |  2                         2             ||             /    ____           \                         |
 | x  - x - 3                               ||   ____      |2*\/ 13 *(-1/2 + x)|                         |
 |                                          ||-\/ 13 *atanh|-------------------|                         |
/                                           ||             \         13        /                 2       |
                                            ||-----------------------------------  for (-1/2 + x)  < 13/4|
                                            \\                 26                                        /
∫x(x2−x)−3 dx=C+2({−13acoth⁡(213(x−12)13)26for (x−12)2>134−13atanh⁡(213(x−12)13)26for (x−12)2<134)+log⁡(x2−x−3)2\int \frac{x}{\left(x^{2} - x\right) - 3}\, dx = C + 2 \left(\begin{cases} - \frac{\sqrt{13} \operatorname{acoth}{\left(\frac{2 \sqrt{13} \left(x - \frac{1}{2}\right)}{13} \right)}}{26} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} > \frac{13}{4} \\- \frac{\sqrt{13} \operatorname{atanh}{\left(\frac{2 \sqrt{13} \left(x - \frac{1}{2}\right)}{13} \right)}}{26} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} < \frac{13}{4} \end{cases}\right) + \frac{\log{\left(x^{2} - x - 3 \right)}}{2}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.5-0.5
Trazar el gráfico f(x)
Respuesta [src] 
/      ____\    /      ____\   /      ____\ /          /        ____\\   /      ____\    /        ____\   /      ____\ /          /      ____\\
|1   \/ 13 |    |1   \/ 13 |   |1   \/ 13 | |          |  1   \/ 13 ||   |1   \/ 13 |    |  1   \/ 13 |   |1   \/ 13 | |          |1   \/ 13 ||
|- - ------|*log|- + ------| + |- + ------|*|pi*I + log|- - + ------|| - |- - ------|*log|- - + ------| - |- + ------|*|pi*I + log|- + ------||
\2     26  /    \2     2   /   \2     26  / \          \  2     2   //   \2     26  /    \  2     2   /   \2     26  / \          \2     2   //
−(12−1326)log⁡(−12+132)+(12−1326)log⁡(12+132)−(1326+12)(log⁡(12+132)+iπ)+(1326+12)(log⁡(−12+132)+iπ)- \left(\frac{1}{2} - \frac{\sqrt{13}}{26}\right) \log{\left(- \frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{13}}{26}\right) \log{\left(\frac{1}{2} + \frac{\sqrt{13}}{2} \right)} - \left(\frac{\sqrt{13}}{26} + \frac{1}{2}\right) \left(\log{\left(\frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right) + \left(\frac{\sqrt{13}}{26} + \frac{1}{2}\right) \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right) 
=
= 
/      ____\    /      ____\   /      ____\ /          /        ____\\   /      ____\    /        ____\   /      ____\ /          /      ____\\
|1   \/ 13 |    |1   \/ 13 |   |1   \/ 13 | |          |  1   \/ 13 ||   |1   \/ 13 |    |  1   \/ 13 |   |1   \/ 13 | |          |1   \/ 13 ||
|- - ------|*log|- + ------| + |- + ------|*|pi*I + log|- - + ------|| - |- - ------|*log|- - + ------| - |- + ------|*|pi*I + log|- + ------||
\2     26  /    \2     2   /   \2     26  / \          \  2     2   //   \2     26  /    \  2     2   /   \2     26  / \          \2     2   //
−(12−1326)log⁡(−12+132)+(12−1326)log⁡(12+132)−(1326+12)(log⁡(12+132)+iπ)+(1326+12)(log⁡(−12+132)+iπ)- \left(\frac{1}{2} - \frac{\sqrt{13}}{26}\right) \log{\left(- \frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{13}}{26}\right) \log{\left(\frac{1}{2} + \frac{\sqrt{13}}{2} \right)} - \left(\frac{\sqrt{13}}{26} + \frac{1}{2}\right) \left(\log{\left(\frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right) + \left(\frac{\sqrt{13}}{26} + \frac{1}{2}\right) \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right) 
(1/2 - sqrt(13)/26)*log(1/2 + sqrt(13)/2) + (1/2 + sqrt(13)/26)*(pi*i + log(-1/2 + sqrt(13)/2)) - (1/2 - sqrt(13)/26)*log(-1/2 + sqrt(13)/2) - (1/2 + sqrt(13)/26)*(pi*i + log(1/2 + sqrt(13)/2))
Respuesta numérica [src] 
-0.157983635931898
-0.157983635931898

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

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