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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |      2 + x        
 |  -------------- dx
 |     2             
 |  - x  - 5*x - 1   
 |                   
/                    
0
$$\int\limits_{0}^{1} \frac{x + 2}{\left(- x^{2} - 5 x\right) - 1}\, dx$$ 
Integral((2 + x)/(-x^2 - 5*x - 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                             //             /    ____          \                        \                     
                             ||   ____      |2*\/ 21 *(5/2 + x)|                        |                     
                             ||-\/ 21 *acoth|------------------|                        |                     
  /                          ||             \        21        /                2       |                     
 |                           ||----------------------------------  for (5/2 + x)  > 21/4|      /      2      \
 |     2 + x                 ||                42                                       |   log\-1 - x  - 5*x/
 | -------------- dx = C + 2*|<                                                         | - ------------------
 |    2                      ||             /    ____          \                        |           2         
 | - x  - 5*x - 1            ||   ____      |2*\/ 21 *(5/2 + x)|                        |                     
 |                           ||-\/ 21 *atanh|------------------|                        |                     
/                            ||             \        21        /                2       |                     
                             ||----------------------------------  for (5/2 + x)  < 21/4|                     
                             \\                42                                       /
$$\int \frac{x + 2}{\left(- x^{2} - 5 x\right) - 1}\, dx = C + 2 \left(\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left(\frac{2 \sqrt{21} \left(x + \frac{5}{2}\right)}{21} \right)}}{42} & \text{for}\: \left(x + \frac{5}{2}\right)^{2} > \frac{21}{4} \\- \frac{\sqrt{21} \operatorname{atanh}{\left(\frac{2 \sqrt{21} \left(x + \frac{5}{2}\right)}{21} \right)}}{42} & \text{for}\: \left(x + \frac{5}{2}\right)^{2} < \frac{21}{4} \end{cases}\right) - \frac{\log{\left(- x^{2} - 5 x - 1 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
/      ____\    /      ____\   /      ____\    /      ____\   /      ____\    /      ____\   /      ____\    /      ____\
|1   \/ 21 |    |5   \/ 21 |   |1   \/ 21 |    |5   \/ 21 |   |1   \/ 21 |    |7   \/ 21 |   |1   \/ 21 |    |7   \/ 21 |
|- - ------|*log|- - ------| + |- + ------|*log|- + ------| - |- - ------|*log|- - ------| - |- + ------|*log|- + ------|
\2     42  /    \2     2   /   \2     42  /    \2     2   /   \2     42  /    \2     2   /   \2     42  /    \2     2   /
$$- \left(\frac{\sqrt{21}}{42} + \frac{1}{2}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{7}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{21}}{42}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{21}}{2} \right)} - \left(\frac{1}{2} - \frac{\sqrt{21}}{42}\right) \log{\left(\frac{7}{2} - \frac{\sqrt{21}}{2} \right)} + \left(\frac{\sqrt{21}}{42} + \frac{1}{2}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{5}{2} \right)}$$ 
=
= 
/      ____\    /      ____\   /      ____\    /      ____\   /      ____\    /      ____\   /      ____\    /      ____\
|1   \/ 21 |    |5   \/ 21 |   |1   \/ 21 |    |5   \/ 21 |   |1   \/ 21 |    |7   \/ 21 |   |1   \/ 21 |    |7   \/ 21 |
|- - ------|*log|- - ------| + |- + ------|*log|- + ------| - |- - ------|*log|- - ------| - |- + ------|*log|- + ------|
\2     42  /    \2     2   /   \2     42  /    \2     2   /   \2     42  /    \2     2   /   \2     42  /    \2     2   /
$$- \left(\frac{\sqrt{21}}{42} + \frac{1}{2}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{7}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{21}}{42}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{21}}{2} \right)} - \left(\frac{1}{2} - \frac{\sqrt{21}}{42}\right) \log{\left(\frac{7}{2} - \frac{\sqrt{21}}{2} \right)} + \left(\frac{\sqrt{21}}{42} + \frac{1}{2}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{5}{2} \right)}$$ 
(1/2 - sqrt(21)/42)*log(5/2 - sqrt(21)/2) + (1/2 + sqrt(21)/42)*log(5/2 + sqrt(21)/2) - (1/2 - sqrt(21)/42)*log(7/2 - sqrt(21)/2) - (1/2 + sqrt(21)/42)*log(7/2 + sqrt(21)/2)
Respuesta numérica [src] 
-0.802003262569915
-0.802003262569915

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

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