Sr Examen
Integral de x/(x^2-x-3) dx
Solución
Ha introducido
[src]
1 / | | x | ---------- dx | 2 | x - x - 3 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - x\right) - 3}\, dx$$
Integral(x/(x^2 - x - 3), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ____ \ \
|| ____ |2*\/ 13 *(-1/2 + x)| |
||-\/ 13 *acoth|-------------------| |
/ || \ 13 / 2 |
| / 2 \ ||----------------------------------- for (-1/2 + x) > 13/4|
| x log\-3 + x - x/ || 26 |
| ---------- dx = C + ---------------- + 2*|< |
| 2 2 || / ____ \ |
| x - x - 3 || ____ |2*\/ 13 *(-1/2 + x)| |
| ||-\/ 13 *atanh|-------------------| |
/ || \ 13 / 2 |
||----------------------------------- for (-1/2 + x) < 13/4|
\\ 26 /
$$\int \frac{x}{\left(x^{2} - x\right) - 3}\, dx = C + 2 \left(\begin{cases} - \frac{\sqrt{13} \operatorname{acoth}{\left(\frac{2 \sqrt{13} \left(x - \frac{1}{2}\right)}{13} \right)}}{26} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} > \frac{13}{4} \\- \frac{\sqrt{13} \operatorname{atanh}{\left(\frac{2 \sqrt{13} \left(x - \frac{1}{2}\right)}{13} \right)}}{26} & \text{for}\: \left(x - \frac{1}{2}\right)^{2} < \frac{13}{4} \end{cases}\right) + \frac{\log{\left(x^{2} - x - 3 \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\ / ____\ / / ____\\ / ____\ / ____\ / ____\ / / ____\\ |1 \/ 13 | |1 \/ 13 | |1 \/ 13 | | | 1 \/ 13 || |1 \/ 13 | | 1 \/ 13 | |1 \/ 13 | | |1 \/ 13 || |- - ------|*log|- + ------| + |- + ------|*|pi*I + log|- - + ------|| - |- - ------|*log|- - + ------| - |- + ------|*|pi*I + log|- + ------|| \2 26 / \2 2 / \2 26 / \ \ 2 2 // \2 26 / \ 2 2 / \2 26 / \ \2 2 //
$$- \left(\frac{1}{2} - \frac{\sqrt{13}}{26}\right) \log{\left(- \frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{13}}{26}\right) \log{\left(\frac{1}{2} + \frac{\sqrt{13}}{2} \right)} - \left(\frac{\sqrt{13}}{26} + \frac{1}{2}\right) \left(\log{\left(\frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right) + \left(\frac{\sqrt{13}}{26} + \frac{1}{2}\right) \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right)$$
=
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/ ____\ / ____\ / ____\ / / ____\\ / ____\ / ____\ / ____\ / / ____\\ |1 \/ 13 | |1 \/ 13 | |1 \/ 13 | | | 1 \/ 13 || |1 \/ 13 | | 1 \/ 13 | |1 \/ 13 | | |1 \/ 13 || |- - ------|*log|- + ------| + |- + ------|*|pi*I + log|- - + ------|| - |- - ------|*log|- - + ------| - |- + ------|*|pi*I + log|- + ------|| \2 26 / \2 2 / \2 26 / \ \ 2 2 // \2 26 / \ 2 2 / \2 26 / \ \2 2 //
$$- \left(\frac{1}{2} - \frac{\sqrt{13}}{26}\right) \log{\left(- \frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{13}}{26}\right) \log{\left(\frac{1}{2} + \frac{\sqrt{13}}{2} \right)} - \left(\frac{\sqrt{13}}{26} + \frac{1}{2}\right) \left(\log{\left(\frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right) + \left(\frac{\sqrt{13}}{26} + \frac{1}{2}\right) \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{13}}{2} \right)} + i \pi\right)$$
(1/2 - sqrt(13)/26)*log(1/2 + sqrt(13)/2) + (1/2 + sqrt(13)/26)*(pi*i + log(-1/2 + sqrt(13)/2)) - (1/2 - sqrt(13)/26)*log(-1/2 + sqrt(13)/2) - (1/2 + sqrt(13)/26)*(pi*i + log(1/2 + sqrt(13)/2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.