Sr Examen
Integral de x/(x^2-2x-2) dx
Solución
Ha introducido
[src]
1 / | | x | ------------ dx | 2 | x - 2*x - 2 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - 2 x\right) - 2}\, dx$$
Integral(x/(x^2 - 2*x - 2), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ___ \ \
|| ___ |\/ 3 *(-1 + x)| |
||-\/ 3 *acoth|--------------| |
/ || \ 3 / 2 |
| / 2 \ ||----------------------------- for (-1 + x) > 3|
| x log\-2 + x - 2*x/ || 3 |
| ------------ dx = C + ------------------ + |< |
| 2 2 || / ___ \ |
| x - 2*x - 2 || ___ |\/ 3 *(-1 + x)| |
| ||-\/ 3 *atanh|--------------| |
/ || \ 3 / 2 |
||----------------------------- for (-1 + x) < 3|
\\ 3 /
$$\int \frac{x}{\left(x^{2} - 2 x\right) - 2}\, dx = C + \begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} \left(x - 1\right)}{3} \right)}}{3} & \text{for}\: \left(x - 1\right)^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} \left(x - 1\right)}{3} \right)}}{3} & \text{for}\: \left(x - 1\right)^{2} < 3 \end{cases} + \frac{\log{\left(x^{2} - 2 x - 2 \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\ / ___\ / ___\ |1 \/ 3 | / ___\ |1 \/ 3 | / / ___\\ |1 \/ 3 | / ___\ |1 \/ 3 | / / ___\\ |- - -----|*log\\/ 3 / + |- + -----|*\pi*I + log\\/ 3 // - |- - -----|*log\-1 + \/ 3 / - |- + -----|*\pi*I + log\1 + \/ 3 // \2 6 / \2 6 / \2 6 / \2 6 /
$$- \left(\frac{1}{2} - \frac{\sqrt{3}}{6}\right) \log{\left(-1 + \sqrt{3} \right)} + \left(\frac{1}{2} - \frac{\sqrt{3}}{6}\right) \log{\left(\sqrt{3} \right)} - \left(\frac{\sqrt{3}}{6} + \frac{1}{2}\right) \left(\log{\left(1 + \sqrt{3} \right)} + i \pi\right) + \left(\frac{\sqrt{3}}{6} + \frac{1}{2}\right) \left(\log{\left(\sqrt{3} \right)} + i \pi\right)$$
=
=
/ ___\ / ___\ / ___\ / ___\ |1 \/ 3 | / ___\ |1 \/ 3 | / / ___\\ |1 \/ 3 | / ___\ |1 \/ 3 | / / ___\\ |- - -----|*log\\/ 3 / + |- + -----|*\pi*I + log\\/ 3 // - |- - -----|*log\-1 + \/ 3 / - |- + -----|*\pi*I + log\1 + \/ 3 // \2 6 / \2 6 / \2 6 / \2 6 /
$$- \left(\frac{1}{2} - \frac{\sqrt{3}}{6}\right) \log{\left(-1 + \sqrt{3} \right)} + \left(\frac{1}{2} - \frac{\sqrt{3}}{6}\right) \log{\left(\sqrt{3} \right)} - \left(\frac{\sqrt{3}}{6} + \frac{1}{2}\right) \left(\log{\left(1 + \sqrt{3} \right)} + i \pi\right) + \left(\frac{\sqrt{3}}{6} + \frac{1}{2}\right) \left(\log{\left(\sqrt{3} \right)} + i \pi\right)$$
(1/2 - sqrt(3)/6)*log(sqrt(3)) + (1/2 + sqrt(3)/6)*(pi*i + log(sqrt(3))) - (1/2 - sqrt(3)/6)*log(-1 + sqrt(3)) - (1/2 + sqrt(3)/6)*(pi*i + log(1 + sqrt(3)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.