Sr Examen
Integral de (x+3)/(x^2-2x-5) dx
Solución
Ha introducido
[src]
1 / | | x + 3 | ------------ dx | 2 | x - 2*x - 5 | / 0
$$\int\limits_{0}^{1} \frac{x + 3}{\left(x^{2} - 2 x\right) - 5}\, dx$$
Integral((x + 3)/(x^2 - 2*x - 5), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ___ \ \
|| ___ |\/ 6 *(-1 + x)| |
||-\/ 6 *acoth|--------------| |
/ || \ 6 / 2 |
| / 2 \ ||----------------------------- for (-1 + x) > 6|
| x + 3 log\-5 + x - 2*x/ || 6 |
| ------------ dx = C + ------------------ + 4*|< |
| 2 2 || / ___ \ |
| x - 2*x - 5 || ___ |\/ 6 *(-1 + x)| |
| ||-\/ 6 *atanh|--------------| |
/ || \ 6 / 2 |
||----------------------------- for (-1 + x) < 6|
\\ 6 /
$$\int \frac{x + 3}{\left(x^{2} - 2 x\right) - 5}\, dx = C + 4 \left(\begin{cases} - \frac{\sqrt{6} \operatorname{acoth}{\left(\frac{\sqrt{6} \left(x - 1\right)}{6} \right)}}{6} & \text{for}\: \left(x - 1\right)^{2} > 6 \\- \frac{\sqrt{6} \operatorname{atanh}{\left(\frac{\sqrt{6} \left(x - 1\right)}{6} \right)}}{6} & \text{for}\: \left(x - 1\right)^{2} < 6 \end{cases}\right) + \frac{\log{\left(x^{2} - 2 x - 5 \right)}}{2}$$
Gráfica
Respuesta
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/ ___\ / ___\ / ___\ / ___\ |1 \/ 6 | / ___\ |1 \/ 6 | / / ___\\ |1 \/ 6 | / ___\ |1 \/ 6 | / / ___\\ |- - -----|*log\\/ 6 / + |- + -----|*\pi*I + log\\/ 6 // - |- - -----|*log\-1 + \/ 6 / - |- + -----|*\pi*I + log\1 + \/ 6 // \2 3 / \2 3 / \2 3 / \2 3 /
$$\left(\frac{1}{2} - \frac{\sqrt{6}}{3}\right) \log{\left(\sqrt{6} \right)} - \left(\frac{1}{2} - \frac{\sqrt{6}}{3}\right) \log{\left(-1 + \sqrt{6} \right)} - \left(\frac{1}{2} + \frac{\sqrt{6}}{3}\right) \left(\log{\left(1 + \sqrt{6} \right)} + i \pi\right) + \left(\frac{1}{2} + \frac{\sqrt{6}}{3}\right) \left(\log{\left(\sqrt{6} \right)} + i \pi\right)$$
=
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/ ___\ / ___\ / ___\ / ___\ |1 \/ 6 | / ___\ |1 \/ 6 | / / ___\\ |1 \/ 6 | / ___\ |1 \/ 6 | / / ___\\ |- - -----|*log\\/ 6 / + |- + -----|*\pi*I + log\\/ 6 // - |- - -----|*log\-1 + \/ 6 / - |- + -----|*\pi*I + log\1 + \/ 6 // \2 3 / \2 3 / \2 3 / \2 3 /
$$\left(\frac{1}{2} - \frac{\sqrt{6}}{3}\right) \log{\left(\sqrt{6} \right)} - \left(\frac{1}{2} - \frac{\sqrt{6}}{3}\right) \log{\left(-1 + \sqrt{6} \right)} - \left(\frac{1}{2} + \frac{\sqrt{6}}{3}\right) \left(\log{\left(1 + \sqrt{6} \right)} + i \pi\right) + \left(\frac{1}{2} + \frac{\sqrt{6}}{3}\right) \left(\log{\left(\sqrt{6} \right)} + i \pi\right)$$
(1/2 - sqrt(6)/3)*log(sqrt(6)) + (1/2 + sqrt(6)/3)*(pi*i + log(sqrt(6))) - (1/2 - sqrt(6)/3)*log(-1 + sqrt(6)) - (1/2 + sqrt(6)/3)*(pi*i + log(1 + sqrt(6)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.