Sr Examen
Integral de 1/(8*x^2-9) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------- dx | 2 | 8*x - 9 | / 0
$$\int\limits_{0}^{1} \frac{1}{8 x^{2} - 9}\, dx$$
Integral(1/(8*x^2 - 9), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=8, c=-9, context=1/(8*x**2 - 9), symbol=x), False), (ArccothRule(a=1, b=8, c=-9, context=1/(8*x**2 - 9), symbol=x), x**2 > 9/8), (ArctanhRule(a=1, b=8, c=-9, context=1/(8*x**2 - 9), symbol=x), x**2 < 9/8)], context=1/(8*x**2 - 9), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ___\ \
|| ___ |2*x*\/ 2 | |
||-\/ 2 *acoth|---------| |
/ || \ 3 / 2 |
| ||------------------------ for x > 9/8|
| 1 || 12 |
| -------- dx = C + |< |
| 2 || / ___\ |
| 8*x - 9 || ___ |2*x*\/ 2 | |
| ||-\/ 2 *atanh|---------| |
/ || \ 3 / 2 |
||------------------------ for x < 9/8|
\\ 12 /
$$\int \frac{1}{8 x^{2} - 9}\, dx = C + \begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{2 \sqrt{2} x}{3} \right)}}{12} & \text{for}\: x^{2} > \frac{9}{8} \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{2 \sqrt{2} x}{3} \right)}}{12} & \text{for}\: x^{2} < \frac{9}{8} \end{cases}$$
Gráfica
Respuesta
[src]
/ / ___\\ / ___\ / / ___\\ / ___\
___ | |3*\/ 2 || ___ | 3*\/ 2 | ___ | | 3*\/ 2 || ___ |3*\/ 2 |
\/ 2 *|pi*I + log|-------|| \/ 2 *log|1 + -------| \/ 2 *|pi*I + log|-1 + -------|| \/ 2 *log|-------|
\ \ 4 // \ 4 / \ \ 4 // \ 4 /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
24 24 24 24
$$- \frac{\sqrt{2} \log{\left(1 + \frac{3 \sqrt{2}}{4} \right)}}{24} + \frac{\sqrt{2} \log{\left(\frac{3 \sqrt{2}}{4} \right)}}{24} - \frac{\sqrt{2} \left(\log{\left(\frac{3 \sqrt{2}}{4} \right)} + i \pi\right)}{24} + \frac{\sqrt{2} \left(\log{\left(-1 + \frac{3 \sqrt{2}}{4} \right)} + i \pi\right)}{24}$$
=
=
/ / ___\\ / ___\ / / ___\\ / ___\
___ | |3*\/ 2 || ___ | 3*\/ 2 | ___ | | 3*\/ 2 || ___ |3*\/ 2 |
\/ 2 *|pi*I + log|-------|| \/ 2 *log|1 + -------| \/ 2 *|pi*I + log|-1 + -------|| \/ 2 *log|-------|
\ \ 4 // \ 4 / \ \ 4 // \ 4 /
- --------------------------- - ---------------------- + -------------------------------- + ------------------
24 24 24 24
$$- \frac{\sqrt{2} \log{\left(1 + \frac{3 \sqrt{2}}{4} \right)}}{24} + \frac{\sqrt{2} \log{\left(\frac{3 \sqrt{2}}{4} \right)}}{24} - \frac{\sqrt{2} \left(\log{\left(\frac{3 \sqrt{2}}{4} \right)} + i \pi\right)}{24} + \frac{\sqrt{2} \left(\log{\left(-1 + \frac{3 \sqrt{2}}{4} \right)} + i \pi\right)}{24}$$
-sqrt(2)*(pi*i + log(3*sqrt(2)/4))/24 - sqrt(2)*log(1 + 3*sqrt(2)/4)/24 + sqrt(2)*(pi*i + log(-1 + 3*sqrt(2)/4))/24 + sqrt(2)*log(3*sqrt(2)/4)/24
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.