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