Sr Examen
Integral de dx/(x^2-x-1) dx
Solución
Ha introducido
[src]
1 / | | 1 | ---------- dx | 2 | x - x - 1 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} - x\right) - 1}\, dx$$
Integral(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|
| 1 || 10 |
| ---------- dx = C + 4*|< |
| 2 || / ___ \ |
| x - x - 1 || ___ |2*\/ 5 *(-1/2 + x)| |
| ||-\/ 5 *atanh|------------------| |
/ || \ 5 / 2 |
||--------------------------------- for (-1/2 + x) < 5/4|
\\ 10 /
$$\int \frac{1}{\left(x^{2} - x\right) - 1}\, dx = C + 4 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right)$$
Gráfica
Respuesta
[src]
/ / ___\\ / ___\ / / ___\\ / ___\
___ | |1 \/ 5 || ___ |1 \/ 5 | ___ | | 1 \/ 5 || ___ | 1 \/ 5 |
\/ 5 *|pi*I + log|- + -----|| \/ 5 *log|- + -----| \/ 5 *|pi*I + log|- - + -----|| \/ 5 *log|- - + -----|
\ \2 2 // \2 2 / \ \ 2 2 // \ 2 2 /
- ----------------------------- - -------------------- + ------------------------------- + ----------------------
5 5 5 5
$$- \frac{\sqrt{5} \log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{5} + \frac{\sqrt{5} \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{5} - \frac{\sqrt{5} \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{5} + \frac{\sqrt{5} \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{5}$$
=
=
/ / ___\\ / ___\ / / ___\\ / ___\
___ | |1 \/ 5 || ___ |1 \/ 5 | ___ | | 1 \/ 5 || ___ | 1 \/ 5 |
\/ 5 *|pi*I + log|- + -----|| \/ 5 *log|- + -----| \/ 5 *|pi*I + log|- - + -----|| \/ 5 *log|- - + -----|
\ \2 2 // \2 2 / \ \ 2 2 // \ 2 2 /
- ----------------------------- - -------------------- + ------------------------------- + ----------------------
5 5 5 5
$$- \frac{\sqrt{5} \log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{5} + \frac{\sqrt{5} \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{5} - \frac{\sqrt{5} \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{5} + \frac{\sqrt{5} \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{5}$$
-sqrt(5)*(pi*i + log(1/2 + sqrt(5)/2))/5 - sqrt(5)*log(1/2 + sqrt(5)/2)/5 + sqrt(5)*(pi*i + log(-1/2 + sqrt(5)/2))/5 + sqrt(5)*log(-1/2 + sqrt(5)/2)/5
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.