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