Sr Examen
Integral de dx/(1-(2/3x^2)) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------- dx | 2 | 2*x | 1 - ---- | 3 | / 0
$$\int\limits_{0}^{1} \frac{1}{1 - \frac{2 x^{2}}{3}}\, dx$$
Integral(1/(1 - 2*x^2/3), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-2/3, c=1, context=1/(1 - 2*x**2/3), symbol=x), False), (ArccothRule(a=1, b=-2/3, c=1, context=1/(1 - 2*x**2/3), symbol=x), x**2 > 3/2), (ArctanhRule(a=1, b=-2/3, c=1, context=1/(1 - 2*x**2/3), symbol=x), x**2 < 3/2)], context=1/(1 - 2*x**2/3), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ___\ \
|| ___ |x*\/ 6 | |
||\/ 6 *acoth|-------| |
/ || \ 3 / 2 |
| ||-------------------- for x > 3/2|
| 1 || 2 |
| -------- dx = C + |< |
| 2 || / ___\ |
| 2*x || ___ |x*\/ 6 | |
| 1 - ---- ||\/ 6 *atanh|-------| |
| 3 || \ 3 / 2 |
| ||-------------------- for x < 3/2|
/ \\ 2 /
$$\int \frac{1}{1 - \frac{2 x^{2}}{3}}\, dx = C + \begin{cases} \frac{\sqrt{6} \operatorname{acoth}{\left(\frac{\sqrt{6} x}{3} \right)}}{2} & \text{for}\: x^{2} > \frac{3}{2} \\\frac{\sqrt{6} \operatorname{atanh}{\left(\frac{\sqrt{6} x}{3} \right)}}{2} & \text{for}\: x^{2} < \frac{3}{2} \end{cases}$$
Gráfica
Respuesta
[src]
/ / ___\\ / ___\ / / ___\\ / ___\
___ | | \/ 6 || ___ |\/ 6 | ___ | |\/ 6 || ___ | \/ 6 |
\/ 6 *|pi*I + log|-1 + -----|| \/ 6 *log|-----| \/ 6 *|pi*I + log|-----|| \/ 6 *log|1 + -----|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
4 4 4 4
$$- \frac{\sqrt{6} \log{\left(\frac{\sqrt{6}}{2} \right)}}{4} + \frac{\sqrt{6} \log{\left(1 + \frac{\sqrt{6}}{2} \right)}}{4} - \frac{\sqrt{6} \left(\log{\left(-1 + \frac{\sqrt{6}}{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{6} \left(\log{\left(\frac{\sqrt{6}}{2} \right)} + i \pi\right)}{4}$$
=
=
/ / ___\\ / ___\ / / ___\\ / ___\
___ | | \/ 6 || ___ |\/ 6 | ___ | |\/ 6 || ___ | \/ 6 |
\/ 6 *|pi*I + log|-1 + -----|| \/ 6 *log|-----| \/ 6 *|pi*I + log|-----|| \/ 6 *log|1 + -----|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
4 4 4 4
$$- \frac{\sqrt{6} \log{\left(\frac{\sqrt{6}}{2} \right)}}{4} + \frac{\sqrt{6} \log{\left(1 + \frac{\sqrt{6}}{2} \right)}}{4} - \frac{\sqrt{6} \left(\log{\left(-1 + \frac{\sqrt{6}}{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{6} \left(\log{\left(\frac{\sqrt{6}}{2} \right)} + i \pi\right)}{4}$$
-sqrt(6)*(pi*i + log(-1 + sqrt(6)/2))/4 - sqrt(6)*log(sqrt(6)/2)/4 + sqrt(6)*(pi*i + log(sqrt(6)/2))/4 + sqrt(6)*log(1 + sqrt(6)/2)/4
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.