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Resolución de integrales/ x^2-x/ (3x-1)/ (3x-1)/(x^2-x-1)

Integral de (3x-1)/(x^2-x-1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1              
  /              
 |               
 |   3*x - 1     
 |  ---------- dx
 |   2           
 |  x  - x - 1   
 |               
/                
0
$$\int\limits_{0}^{1} \frac{3 x - 1}{\left(x^{2} - x\right) - 1}\, dx$$ 
Integral((3*x - 1)/(x^2 - x - 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                         //            /    ___           \                        \                     
                         ||   ___      |2*\/ 5 *(-1/2 + x)|                        |                     
                         ||-\/ 5 *acoth|------------------|                        |                     
  /                      ||            \        5         /                 2      |                     
 |                       ||---------------------------------  for (-1/2 + x)  > 5/4|        /      2    \
 |  3*x - 1              ||                10                                      |   3*log\-1 + x  - x/
 | ---------- dx = C + 2*|<                                                        | + ------------------
 |  2                    ||            /    ___           \                        |           2         
 | x  - x - 1            ||   ___      |2*\/ 5 *(-1/2 + x)|                        |                     
 |                       ||-\/ 5 *atanh|------------------|                        |                     
/                        ||            \        5         /                 2      |                     
                         ||---------------------------------  for (-1/2 + x)  < 5/4|                     
                         \\                10                                      /
$$\int \frac{3 x - 1}{\left(x^{2} - x\right) - 1}\, dx = C + 2 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) + \frac{3 \log{\left(x^{2} - x - 1 \right)}}{2}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
/      ___\    /      ___\   /      ___\ /          /        ___\\   /      ___\    /        ___\   /      ___\ /          /      ___\\
|3   \/ 5 |    |1   \/ 5 |   |3   \/ 5 | |          |  1   \/ 5 ||   |3   \/ 5 |    |  1   \/ 5 |   |3   \/ 5 | |          |1   \/ 5 ||
|- - -----|*log|- + -----| + |- + -----|*|pi*I + log|- - + -----|| - |- - -----|*log|- - + -----| - |- + -----|*|pi*I + log|- + -----||
\2     10 /    \2     2  /   \2     10 / \          \  2     2  //   \2     10 /    \  2     2  /   \2     10 / \          \2     2  //
$$\left(\frac{3}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{3}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{\sqrt{5}}{10} + \frac{3}{2}\right) \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) + \left(\frac{\sqrt{5}}{10} + \frac{3}{2}\right) \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)$$ 
=
= 
/      ___\    /      ___\   /      ___\ /          /        ___\\   /      ___\    /        ___\   /      ___\ /          /      ___\\
|3   \/ 5 |    |1   \/ 5 |   |3   \/ 5 | |          |  1   \/ 5 ||   |3   \/ 5 |    |  1   \/ 5 |   |3   \/ 5 | |          |1   \/ 5 ||
|- - -----|*log|- + -----| + |- + -----|*|pi*I + log|- - + -----|| - |- - -----|*log|- - + -----| - |- + -----|*|pi*I + log|- + -----||
\2     10 /    \2     2  /   \2     10 / \          \  2     2  //   \2     10 /    \  2     2  /   \2     10 / \          \2     2  //
$$\left(\frac{3}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{3}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} - \left(\frac{\sqrt{5}}{10} + \frac{3}{2}\right) \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) + \left(\frac{\sqrt{5}}{10} + \frac{3}{2}\right) \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)$$ 
(3/2 - sqrt(5)/10)*log(1/2 + sqrt(5)/2) + (3/2 + sqrt(5)/10)*(pi*i + log(-1/2 + sqrt(5)/2)) - (3/2 - sqrt(5)/10)*log(-1/2 + sqrt(5)/2) - (3/2 + sqrt(5)/10)*(pi*i + 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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