Sr Examen
Integral de dx/(5*x^2-16) dx
Solución
Ha introducido
[src]
1 / | | 1 | --------- dx | 2 | 5*x - 16 | / 0
$$\int\limits_{0}^{1} \frac{1}{5 x^{2} - 16}\, dx$$
Integral(1/(5*x^2 - 16), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=5, c=-16, context=1/(5*x**2 - 16), symbol=x), False), (ArccothRule(a=1, b=5, c=-16, context=1/(5*x**2 - 16), symbol=x), x**2 > 16/5), (ArctanhRule(a=1, b=5, c=-16, context=1/(5*x**2 - 16), symbol=x), x**2 < 16/5)], context=1/(5*x**2 - 16), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ___\ \
|| ___ |x*\/ 5 | |
||-\/ 5 *acoth|-------| |
/ || \ 4 / 2 |
| ||---------------------- for x > 16/5|
| 1 || 20 |
| --------- dx = C + |< |
| 2 || / ___\ |
| 5*x - 16 || ___ |x*\/ 5 | |
| ||-\/ 5 *atanh|-------| |
/ || \ 4 / 2 |
||---------------------- for x < 16/5|
\\ 20 /
$$\int \frac{1}{5 x^{2} - 16}\, dx = C + \begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{\sqrt{5} x}{4} \right)}}{20} & \text{for}\: x^{2} > \frac{16}{5} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{\sqrt{5} x}{4} \right)}}{20} & \text{for}\: x^{2} < \frac{16}{5} \end{cases}$$
Gráfica
Respuesta
[src]
/ / ___\\ / ___\ / / ___\\ / ___\
___ | |4*\/ 5 || ___ | 4*\/ 5 | ___ | | 4*\/ 5 || ___ |4*\/ 5 |
\/ 5 *|pi*I + log|-------|| \/ 5 *log|1 + -------| \/ 5 *|pi*I + log|-1 + -------|| \/ 5 *log|-------|
\ \ 5 // \ 5 / \ \ 5 // \ 5 /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
40 40 40 40
$$- \frac{\sqrt{5} \log{\left(1 + \frac{4 \sqrt{5}}{5} \right)}}{40} + \frac{\sqrt{5} \log{\left(\frac{4 \sqrt{5}}{5} \right)}}{40} - \frac{\sqrt{5} \left(\log{\left(\frac{4 \sqrt{5}}{5} \right)} + i \pi\right)}{40} + \frac{\sqrt{5} \left(\log{\left(-1 + \frac{4 \sqrt{5}}{5} \right)} + i \pi\right)}{40}$$
=
=
/ / ___\\ / ___\ / / ___\\ / ___\
___ | |4*\/ 5 || ___ | 4*\/ 5 | ___ | | 4*\/ 5 || ___ |4*\/ 5 |
\/ 5 *|pi*I + log|-------|| \/ 5 *log|1 + -------| \/ 5 *|pi*I + log|-1 + -------|| \/ 5 *log|-------|
\ \ 5 // \ 5 / \ \ 5 // \ 5 /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
40 40 40 40
$$- \frac{\sqrt{5} \log{\left(1 + \frac{4 \sqrt{5}}{5} \right)}}{40} + \frac{\sqrt{5} \log{\left(\frac{4 \sqrt{5}}{5} \right)}}{40} - \frac{\sqrt{5} \left(\log{\left(\frac{4 \sqrt{5}}{5} \right)} + i \pi\right)}{40} + \frac{\sqrt{5} \left(\log{\left(-1 + \frac{4 \sqrt{5}}{5} \right)} + i \pi\right)}{40}$$
-sqrt(5)*(pi*i + log(4*sqrt(5)/5))/40 - sqrt(5)*log(1 + 4*sqrt(5)/5)/40 + sqrt(5)*(pi*i + log(-1 + 4*sqrt(5)/5))/40 + sqrt(5)*log(4*sqrt(5)/5)/40
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.