Sr Examen
Integral de 1/(5-2x^2) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------- dx | 2 | 5 - 2*x | / 0
$$\int\limits_{0}^{1} \frac{1}{5 - 2 x^{2}}\, dx$$
Integral(1/(5 - 2*x^2), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-2, c=5, context=1/(5 - 2*x**2), symbol=x), False), (ArccothRule(a=1, b=-2, c=5, context=1/(5 - 2*x**2), symbol=x), x**2 > 5/2), (ArctanhRule(a=1, b=-2, c=5, context=1/(5 - 2*x**2), symbol=x), x**2 < 5/2)], context=1/(5 - 2*x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ____\ \
|| ____ |x*\/ 10 | |
||\/ 10 *acoth|--------| |
/ || \ 5 / 2 |
| ||---------------------- for x > 5/2|
| 1 || 10 |
| -------- dx = C + |< |
| 2 || / ____\ |
| 5 - 2*x || ____ |x*\/ 10 | |
| ||\/ 10 *atanh|--------| |
/ || \ 5 / 2 |
||---------------------- for x < 5/2|
\\ 10 /
$$\int \frac{1}{5 - 2 x^{2}}\, dx = C + \begin{cases} \frac{\sqrt{10} \operatorname{acoth}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} > \frac{5}{2} \\\frac{\sqrt{10} \operatorname{atanh}{\left(\frac{\sqrt{10} x}{5} \right)}}{10} & \text{for}\: x^{2} < \frac{5}{2} \end{cases}$$
Gráfica
Respuesta
[src]
/ / ____\\ / ____\ / / ____\\ / ____\
____ | | \/ 10 || ____ |\/ 10 | ____ | |\/ 10 || ____ | \/ 10 |
\/ 10 *|pi*I + log|-1 + ------|| \/ 10 *log|------| \/ 10 *|pi*I + log|------|| \/ 10 *log|1 + ------|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
20 20 20 20
$$- \frac{\sqrt{10} \log{\left(\frac{\sqrt{10}}{2} \right)}}{20} + \frac{\sqrt{10} \log{\left(1 + \frac{\sqrt{10}}{2} \right)}}{20} - \frac{\sqrt{10} \left(\log{\left(-1 + \frac{\sqrt{10}}{2} \right)} + i \pi\right)}{20} + \frac{\sqrt{10} \left(\log{\left(\frac{\sqrt{10}}{2} \right)} + i \pi\right)}{20}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____\
____ | | \/ 10 || ____ |\/ 10 | ____ | |\/ 10 || ____ | \/ 10 |
\/ 10 *|pi*I + log|-1 + ------|| \/ 10 *log|------| \/ 10 *|pi*I + log|------|| \/ 10 *log|1 + ------|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- -------------------------------- - ------------------ + --------------------------- + ----------------------
20 20 20 20
$$- \frac{\sqrt{10} \log{\left(\frac{\sqrt{10}}{2} \right)}}{20} + \frac{\sqrt{10} \log{\left(1 + \frac{\sqrt{10}}{2} \right)}}{20} - \frac{\sqrt{10} \left(\log{\left(-1 + \frac{\sqrt{10}}{2} \right)} + i \pi\right)}{20} + \frac{\sqrt{10} \left(\log{\left(\frac{\sqrt{10}}{2} \right)} + i \pi\right)}{20}$$
-sqrt(10)*(pi*i + log(-1 + sqrt(10)/2))/20 - sqrt(10)*log(sqrt(10)/2)/20 + sqrt(10)*(pi*i + log(sqrt(10)/2))/20 + sqrt(10)*log(1 + sqrt(10)/2)/20
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.