Sr Examen
Integral de 1/((-5x^2)+13) dx
Solución
Ha introducido
[src]
1 / | | 1 | ----------- dx | 2 | - 5*x + 13 | / 0
$$\int\limits_{0}^{1} \frac{1}{13 - 5 x^{2}}\, dx$$
Integral(1/(-5*x^2 + 13), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-5, c=13, context=1/(13 - 5*x**2), symbol=x), False), (ArccothRule(a=1, b=-5, c=13, context=1/(13 - 5*x**2), symbol=x), x**2 > 13/5), (ArctanhRule(a=1, b=-5, c=13, context=1/(13 - 5*x**2), symbol=x), x**2 < 13/5)], context=1/(13 - 5*x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ____\ \
|| ____ |x*\/ 65 | |
||\/ 65 *acoth|--------| |
/ || \ 13 / 2 |
| ||---------------------- for x > 13/5|
| 1 || 65 |
| ----------- dx = C + |< |
| 2 || / ____\ |
| - 5*x + 13 || ____ |x*\/ 65 | |
| ||\/ 65 *atanh|--------| |
/ || \ 13 / 2 |
||---------------------- for x < 13/5|
\\ 65 /
$$\int \frac{1}{13 - 5 x^{2}}\, dx = C + \begin{cases} \frac{\sqrt{65} \operatorname{acoth}{\left(\frac{\sqrt{65} x}{13} \right)}}{65} & \text{for}\: x^{2} > \frac{13}{5} \\\frac{\sqrt{65} \operatorname{atanh}{\left(\frac{\sqrt{65} x}{13} \right)}}{65} & \text{for}\: x^{2} < \frac{13}{5} \end{cases}$$
Gráfica
Respuesta
[src]
/ / ____\\ / ____\ / / ____\\ / ____\
____ | | \/ 65 || ____ |\/ 65 | ____ | |\/ 65 || ____ | \/ 65 |
\/ 65 *|pi*I + log|-1 + ------|| \/ 65 *log|------| \/ 65 *|pi*I + log|------|| \/ 65 *log|1 + ------|
\ \ 5 // \ 5 / \ \ 5 // \ 5 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
130 130 130 130
$$- \frac{\sqrt{65} \log{\left(\frac{\sqrt{65}}{5} \right)}}{130} + \frac{\sqrt{65} \log{\left(1 + \frac{\sqrt{65}}{5} \right)}}{130} - \frac{\sqrt{65} \left(\log{\left(-1 + \frac{\sqrt{65}}{5} \right)} + i \pi\right)}{130} + \frac{\sqrt{65} \left(\log{\left(\frac{\sqrt{65}}{5} \right)} + i \pi\right)}{130}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____\
____ | | \/ 65 || ____ |\/ 65 | ____ | |\/ 65 || ____ | \/ 65 |
\/ 65 *|pi*I + log|-1 + ------|| \/ 65 *log|------| \/ 65 *|pi*I + log|------|| \/ 65 *log|1 + ------|
\ \ 5 // \ 5 / \ \ 5 // \ 5 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
130 130 130 130
$$- \frac{\sqrt{65} \log{\left(\frac{\sqrt{65}}{5} \right)}}{130} + \frac{\sqrt{65} \log{\left(1 + \frac{\sqrt{65}}{5} \right)}}{130} - \frac{\sqrt{65} \left(\log{\left(-1 + \frac{\sqrt{65}}{5} \right)} + i \pi\right)}{130} + \frac{\sqrt{65} \left(\log{\left(\frac{\sqrt{65}}{5} \right)} + i \pi\right)}{130}$$
-sqrt(65)*(pi*i + log(-1 + sqrt(65)/5))/130 - sqrt(65)*log(sqrt(65)/5)/130 + sqrt(65)*(pi*i + log(sqrt(65)/5))/130 + sqrt(65)*log(1 + sqrt(65)/5)/130
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.