Sr Examen
Integral de dx/((3x^2)-5^2) dx
Solución
Ha introducido
[src]
1 / | | 1 | --------- dx | 2 | 3*x - 25 | / 0
$$\int\limits_{0}^{1} \frac{1}{3 x^{2} - 25}\, dx$$
Integral(1/(3*x^2 - 25), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=3, c=-25, context=1/(3*x**2 - 25), symbol=x), False), (ArccothRule(a=1, b=3, c=-25, context=1/(3*x**2 - 25), symbol=x), x**2 > 25/3), (ArctanhRule(a=1, b=3, c=-25, context=1/(3*x**2 - 25), symbol=x), x**2 < 25/3)], context=1/(3*x**2 - 25), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ___\ \
|| ___ |x*\/ 3 | |
||-\/ 3 *acoth|-------| |
/ || \ 5 / 2 |
| ||---------------------- for x > 25/3|
| 1 || 15 |
| --------- dx = C + |< |
| 2 || / ___\ |
| 3*x - 25 || ___ |x*\/ 3 | |
| ||-\/ 3 *atanh|-------| |
/ || \ 5 / 2 |
||---------------------- for x < 25/3|
\\ 15 /
$$\int \frac{1}{3 x^{2} - 25}\, dx = C + \begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{5} \right)}}{15} & \text{for}\: x^{2} > \frac{25}{3} \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{5} \right)}}{15} & \text{for}\: x^{2} < \frac{25}{3} \end{cases}$$
Gráfica
Respuesta
[src]
/ / ___\\ / ___\ / / ___\\ / ___\
___ | |5*\/ 3 || ___ | 5*\/ 3 | ___ | | 5*\/ 3 || ___ |5*\/ 3 |
\/ 3 *|pi*I + log|-------|| \/ 3 *log|1 + -------| \/ 3 *|pi*I + log|-1 + -------|| \/ 3 *log|-------|
\ \ 3 // \ 3 / \ \ 3 // \ 3 /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
30 30 30 30
$$- \frac{\sqrt{3} \log{\left(1 + \frac{5 \sqrt{3}}{3} \right)}}{30} + \frac{\sqrt{3} \log{\left(\frac{5 \sqrt{3}}{3} \right)}}{30} - \frac{\sqrt{3} \left(\log{\left(\frac{5 \sqrt{3}}{3} \right)} + i \pi\right)}{30} + \frac{\sqrt{3} \left(\log{\left(-1 + \frac{5 \sqrt{3}}{3} \right)} + i \pi\right)}{30}$$
=
=
/ / ___\\ / ___\ / / ___\\ / ___\
___ | |5*\/ 3 || ___ | 5*\/ 3 | ___ | | 5*\/ 3 || ___ |5*\/ 3 |
\/ 3 *|pi*I + log|-------|| \/ 3 *log|1 + -------| \/ 3 *|pi*I + log|-1 + -------|| \/ 3 *log|-------|
\ \ 3 // \ 3 / \ \ 3 // \ 3 /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
30 30 30 30
$$- \frac{\sqrt{3} \log{\left(1 + \frac{5 \sqrt{3}}{3} \right)}}{30} + \frac{\sqrt{3} \log{\left(\frac{5 \sqrt{3}}{3} \right)}}{30} - \frac{\sqrt{3} \left(\log{\left(\frac{5 \sqrt{3}}{3} \right)} + i \pi\right)}{30} + \frac{\sqrt{3} \left(\log{\left(-1 + \frac{5 \sqrt{3}}{3} \right)} + i \pi\right)}{30}$$
-sqrt(3)*(pi*i + log(5*sqrt(3)/3))/30 - sqrt(3)*log(1 + 5*sqrt(3)/3)/30 + sqrt(3)*(pi*i + log(-1 + 5*sqrt(3)/3))/30 + sqrt(3)*log(5*sqrt(3)/3)/30
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.