Sr Examen
Integral de x/(x^4-x^2-1) dx
Solución
Ha introducido
[src]
___ \/ 3 / | | x | ----------- dx | 4 2 | x - x - 1 | / ___ \/ 2
$$\int\limits_{\sqrt{2}}^{\sqrt{3}} \frac{x}{\left(x^{4} - x^{2}\right) - 1}\, dx$$
Integral(x/(x^4 - x^2 - 1), (x, sqrt(2), sqrt(3)))
Respuesta (Indefinida)
[src]
// / ___ / 1 2\\ \
|| |2*\/ 5 *|- - + x || |
|| ___ | \ 2 /| |
||-\/ 5 *acoth|------------------| 2 |
/ || \ 5 / / 1 2\ |
| ||--------------------------------- for |- - + x | > 5/4|
| x || 10 \ 2 / |
| ----------- dx = C + 2*|< |
| 4 2 || / ___ / 1 2\\ |
| x - x - 1 || |2*\/ 5 *|- - + x || |
| || ___ | \ 2 /| |
/ ||-\/ 5 *atanh|------------------| 2 |
|| \ 5 / / 1 2\ |
||--------------------------------- for |- - + x | < 5/4|
\\ 10 \ 2 / /
$$\int \frac{x}{\left(x^{4} - x^{2}\right) - 1}\, dx = C + 2 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x^{2} - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x^{2} - \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x^{2} - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x^{2} - \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right)$$
Gráfica
Respuesta
[src]
/ ___\ / ___\ / ___\ / ___\
___ |3 \/ 5 | ___ |5 \/ 5 | ___ |3 \/ 5 | ___ |5 \/ 5 |
\/ 5 *log|- - -----| \/ 5 *log|- + -----| \/ 5 *log|- + -----| \/ 5 *log|- - -----|
\2 2 / \2 2 / \2 2 / \2 2 /
- -------------------- - -------------------- + -------------------- + --------------------
10 10 10 10
$$- \frac{\sqrt{5} \log{\left(\frac{\sqrt{5}}{2} + \frac{5}{2} \right)}}{10} + \frac{\sqrt{5} \log{\left(\frac{5}{2} - \frac{\sqrt{5}}{2} \right)}}{10} - \frac{\sqrt{5} \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}}{10} + \frac{\sqrt{5} \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}}{10}$$
=
=
/ ___\ / ___\ / ___\ / ___\
___ |3 \/ 5 | ___ |5 \/ 5 | ___ |3 \/ 5 | ___ |5 \/ 5 |
\/ 5 *log|- - -----| \/ 5 *log|- + -----| \/ 5 *log|- + -----| \/ 5 *log|- - -----|
\2 2 / \2 2 / \2 2 / \2 2 /
- -------------------- - -------------------- + -------------------- + --------------------
10 10 10 10
$$- \frac{\sqrt{5} \log{\left(\frac{\sqrt{5}}{2} + \frac{5}{2} \right)}}{10} + \frac{\sqrt{5} \log{\left(\frac{5}{2} - \frac{\sqrt{5}}{2} \right)}}{10} - \frac{\sqrt{5} \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}}{10} + \frac{\sqrt{5} \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}}{10}$$
-sqrt(5)*log(3/2 - sqrt(5)/2)/10 - sqrt(5)*log(5/2 + sqrt(5)/2)/10 + sqrt(5)*log(3/2 + sqrt(5)/2)/10 + sqrt(5)*log(5/2 - sqrt(5)/2)/10
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.