Sr Examen
Integral de 1/(7-2x^2) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------- dx | 2 | 7 - 2*x | / 0
$$\int\limits_{0}^{1} \frac{1}{7 - 2 x^{2}}\, dx$$
Integral(1/(7 - 2*x^2), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-2, c=7, context=1/(7 - 2*x**2), symbol=x), False), (ArccothRule(a=1, b=-2, c=7, context=1/(7 - 2*x**2), symbol=x), x**2 > 7/2), (ArctanhRule(a=1, b=-2, c=7, context=1/(7 - 2*x**2), symbol=x), x**2 < 7/2)], context=1/(7 - 2*x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ____\ \
|| ____ |x*\/ 14 | |
||\/ 14 *acoth|--------| |
/ || \ 7 / 2 |
| ||---------------------- for x > 7/2|
| 1 || 14 |
| -------- dx = C + |< |
| 2 || / ____\ |
| 7 - 2*x || ____ |x*\/ 14 | |
| ||\/ 14 *atanh|--------| |
/ || \ 7 / 2 |
||---------------------- for x < 7/2|
\\ 14 /
$$\int \frac{1}{7 - 2 x^{2}}\, dx = C + \begin{cases} \frac{\sqrt{14} \operatorname{acoth}{\left(\frac{\sqrt{14} x}{7} \right)}}{14} & \text{for}\: x^{2} > \frac{7}{2} \\\frac{\sqrt{14} \operatorname{atanh}{\left(\frac{\sqrt{14} x}{7} \right)}}{14} & \text{for}\: x^{2} < \frac{7}{2} \end{cases}$$
Gráfica
Respuesta
[src]
/ / ____\\ / ____\ / / ____\\ / ____\
____ | | \/ 14 || ____ |\/ 14 | ____ | |\/ 14 || ____ | \/ 14 |
\/ 14 *|pi*I + log|-1 + ------|| \/ 14 *log|------| \/ 14 *|pi*I + log|------|| \/ 14 *log|1 + ------|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
28 28 28 28
$$- \frac{\sqrt{14} \log{\left(\frac{\sqrt{14}}{2} \right)}}{28} + \frac{\sqrt{14} \log{\left(1 + \frac{\sqrt{14}}{2} \right)}}{28} - \frac{\sqrt{14} \left(\log{\left(-1 + \frac{\sqrt{14}}{2} \right)} + i \pi\right)}{28} + \frac{\sqrt{14} \left(\log{\left(\frac{\sqrt{14}}{2} \right)} + i \pi\right)}{28}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____\
____ | | \/ 14 || ____ |\/ 14 | ____ | |\/ 14 || ____ | \/ 14 |
\/ 14 *|pi*I + log|-1 + ------|| \/ 14 *log|------| \/ 14 *|pi*I + log|------|| \/ 14 *log|1 + ------|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
28 28 28 28
$$- \frac{\sqrt{14} \log{\left(\frac{\sqrt{14}}{2} \right)}}{28} + \frac{\sqrt{14} \log{\left(1 + \frac{\sqrt{14}}{2} \right)}}{28} - \frac{\sqrt{14} \left(\log{\left(-1 + \frac{\sqrt{14}}{2} \right)} + i \pi\right)}{28} + \frac{\sqrt{14} \left(\log{\left(\frac{\sqrt{14}}{2} \right)} + i \pi\right)}{28}$$
-sqrt(14)*(pi*i + log(-1 + sqrt(14)/2))/28 - sqrt(14)*log(sqrt(14)/2)/28 + sqrt(14)*(pi*i + log(sqrt(14)/2))/28 + sqrt(14)*log(1 + sqrt(14)/2)/28
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.