Sr Examen
Integral de (x^2-1)/(x^4-3x^2+1) dx
Solución
Ha introducido
[src]
1 / | | 2 | x - 1 | ------------- dx | 4 2 | x - 3*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{x^{2} - 1}{\left(x^{4} - 3 x^{2}\right) + 1}\, dx$$
Integral((x^2 - 1)/(x^4 - 3*x^2 + 1), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ___ \ \ // / ___ \ \
|| ___ |2*\/ 5 *(1/2 + x)| | || ___ |2*\/ 5 *(-1/2 + x)| |
/ ||-\/ 5 *acoth|-----------------| | ||-\/ 5 *acoth|------------------| |
| || \ 5 / 2 | || \ 5 / 2 |
| 2 ||-------------------------------- for (1/2 + x) > 5/4| ||--------------------------------- for (-1/2 + x) > 5/4|
| x - 1 || 10 | || 10 |
| ------------- dx = C + 2*|< | + 2*|< |
| 4 2 || / ___ \ | || / ___ \ |
| x - 3*x + 1 || ___ |2*\/ 5 *(1/2 + x)| | || ___ |2*\/ 5 *(-1/2 + x)| |
| ||-\/ 5 *atanh|-----------------| | ||-\/ 5 *atanh|------------------| |
/ || \ 5 / 2 | || \ 5 / 2 |
||-------------------------------- for (1/2 + x) < 5/4| ||--------------------------------- for (-1/2 + x) < 5/4|
\\ 10 / \\ 10 /
$$\int \frac{x^{2} - 1}{\left(x^{4} - 3 x^{2}\right) + 1}\, dx = C + 2 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) + 2 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x + \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x + \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right)$$
Gráfica
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.