Sr Examen
Integral de 1/((3x^2)-5) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------- dx | 2 | 3*x - 5 | / 0
$$\int\limits_{0}^{1} \frac{1}{3 x^{2} - 5}\, dx$$
Integral(1/(3*x^2 - 5), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=3, c=-5, context=1/(3*x**2 - 5), symbol=x), False), (ArccothRule(a=1, b=3, c=-5, context=1/(3*x**2 - 5), symbol=x), x**2 > 5/3), (ArctanhRule(a=1, b=3, c=-5, context=1/(3*x**2 - 5), symbol=x), x**2 < 5/3)], context=1/(3*x**2 - 5), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ____\ \
|| ____ |x*\/ 15 | |
||-\/ 15 *acoth|--------| |
/ || \ 5 / 2 |
| ||------------------------ for x > 5/3|
| 1 || 15 |
| -------- dx = C + |< |
| 2 || / ____\ |
| 3*x - 5 || ____ |x*\/ 15 | |
| ||-\/ 15 *atanh|--------| |
/ || \ 5 / 2 |
||------------------------ for x < 5/3|
\\ 15 /
$$\int \frac{1}{3 x^{2} - 5}\, dx = C + \begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left(\frac{\sqrt{15} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left(\frac{\sqrt{15} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{5}{3} \end{cases}$$
Gráfica
Respuesta
[src]
/ / ____\\ / ____\ / / ____\\ / ____\
____ | |\/ 15 || ____ | \/ 15 | ____ | | \/ 15 || ____ |\/ 15 |
\/ 15 *|pi*I + log|------|| \/ 15 *log|1 + ------| \/ 15 *|pi*I + log|-1 + ------|| \/ 15 *log|------|
\ \ 3 // \ 3 / \ \ 3 // \ 3 /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
30 30 30 30
$$- \frac{\sqrt{15} \log{\left(1 + \frac{\sqrt{15}}{3} \right)}}{30} + \frac{\sqrt{15} \log{\left(\frac{\sqrt{15}}{3} \right)}}{30} - \frac{\sqrt{15} \left(\log{\left(\frac{\sqrt{15}}{3} \right)} + i \pi\right)}{30} + \frac{\sqrt{15} \left(\log{\left(-1 + \frac{\sqrt{15}}{3} \right)} + i \pi\right)}{30}$$
=
=
/ / ____\\ / ____\ / / ____\\ / ____\
____ | |\/ 15 || ____ | \/ 15 | ____ | | \/ 15 || ____ |\/ 15 |
\/ 15 *|pi*I + log|------|| \/ 15 *log|1 + ------| \/ 15 *|pi*I + log|-1 + ------|| \/ 15 *log|------|
\ \ 3 // \ 3 / \ \ 3 // \ 3 /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
30 30 30 30
$$- \frac{\sqrt{15} \log{\left(1 + \frac{\sqrt{15}}{3} \right)}}{30} + \frac{\sqrt{15} \log{\left(\frac{\sqrt{15}}{3} \right)}}{30} - \frac{\sqrt{15} \left(\log{\left(\frac{\sqrt{15}}{3} \right)} + i \pi\right)}{30} + \frac{\sqrt{15} \left(\log{\left(-1 + \frac{\sqrt{15}}{3} \right)} + i \pi\right)}{30}$$
-sqrt(15)*(pi*i + log(sqrt(15)/3))/30 - sqrt(15)*log(1 + sqrt(15)/3)/30 + sqrt(15)*(pi*i + log(-1 + sqrt(15)/3))/30 + sqrt(15)*log(sqrt(15)/3)/30
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.