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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1              
  /              
 |               
 |    x - 1      
 |  ---------- dx
 |   2           
 |  x  - x - 1   
 |               
/                
0
$$\int\limits_{0}^{1} \frac{x - 1}{\left(x^{2} - x\right) - 1}\, dx$$ 
Integral((x - 1)/(x^2 - x - 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                                            //            /    ___           \                        \
                                            ||   ___      |2*\/ 5 *(-1/2 + x)|                        |
                                            ||-\/ 5 *acoth|------------------|                        |
  /                                         ||            \        5         /                 2      |
 |                        /      2    \     ||---------------------------------  for (-1/2 + x)  > 5/4|
 |   x - 1             log\-1 + x  - x/     ||                10                                      |
 | ---------- dx = C + ---------------- - 2*|<                                                        |
 |  2                         2             ||            /    ___           \                        |
 | x  - x - 1                               ||   ___      |2*\/ 5 *(-1/2 + x)|                        |
 |                                          ||-\/ 5 *atanh|------------------|                        |
/                                           ||            \        5         /                 2      |
                                            ||---------------------------------  for (-1/2 + x)  < 5/4|
                                            \\                10                                      /
$$\int \frac{x - 1}{\left(x^{2} - x\right) - 1}\, dx = C - 2 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) + \frac{\log{\left(x^{2} - x - 1 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
/      ___\ /          /        ___\\   /      ___\    /      ___\   /      ___\ /          /      ___\\   /      ___\    /        ___\
|1   \/ 5 | |          |  1   \/ 5 ||   |1   \/ 5 |    |1   \/ 5 |   |1   \/ 5 | |          |1   \/ 5 ||   |1   \/ 5 |    |  1   \/ 5 |
|- - -----|*|pi*I + log|- - + -----|| + |- + -----|*log|- + -----| - |- - -----|*|pi*I + log|- + -----|| - |- + -----|*log|- - + -----|
\2     10 / \          \  2     2  //   \2     10 /    \2     2  /   \2     10 / \          \2     2  //   \2     10 /    \  2     2  /
$$\left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) + \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)$$ 
=
= 
/      ___\ /          /        ___\\   /      ___\    /      ___\   /      ___\ /          /      ___\\   /      ___\    /        ___\
|1   \/ 5 | |          |  1   \/ 5 ||   |1   \/ 5 |    |1   \/ 5 |   |1   \/ 5 | |          |1   \/ 5 ||   |1   \/ 5 |    |  1   \/ 5 |
|- - -----|*|pi*I + log|- - + -----|| + |- + -----|*log|- + -----| - |- - -----|*|pi*I + log|- + -----|| - |- + -----|*log|- - + -----|
\2     10 / \          \  2     2  //   \2     10 /    \2     2  /   \2     10 / \          \2     2  //   \2     10 /    \  2     2  /
$$\left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) + \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)$$ 
(1/2 - sqrt(5)/10)*(pi*i + log(-1/2 + sqrt(5)/2)) + (1/2 + sqrt(5)/10)*log(1/2 + sqrt(5)/2) - (1/2 - sqrt(5)/10)*(pi*i + log(1/2 + sqrt(5)/2)) - (1/2 + sqrt(5)/10)*log(-1/2 + sqrt(5)/2)
Respuesta numérica [src] 
0.430408940964004
0.430408940964004

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

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