Sr Examen
Integral de dx/(3-2*x^2) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------- dx | 2 | 3 - 2*x | / 0
$$\int\limits_{0}^{1} \frac{1}{3 - 2 x^{2}}\, dx$$
Integral(1/(3 - 2*x^2), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=-2, c=3, context=1/(3 - 2*x**2), symbol=x), False), (ArccothRule(a=1, b=-2, c=3, context=1/(3 - 2*x**2), symbol=x), x**2 > 3/2), (ArctanhRule(a=1, b=-2, c=3, context=1/(3 - 2*x**2), symbol=x), x**2 < 3/2)], context=1/(3 - 2*x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ___\ \
|| ___ |x*\/ 6 | |
||\/ 6 *acoth|-------| |
/ || \ 3 / 2 |
| ||-------------------- for x > 3/2|
| 1 || 6 |
| -------- dx = C + |< |
| 2 || / ___\ |
| 3 - 2*x || ___ |x*\/ 6 | |
| ||\/ 6 *atanh|-------| |
/ || \ 3 / 2 |
||-------------------- for x < 3/2|
\\ 6 /
$$\int \frac{1}{3 - 2 x^{2}}\, dx = C + \begin{cases} \frac{\sqrt{6} \operatorname{acoth}{\left(\frac{\sqrt{6} x}{3} \right)}}{6} & \text{for}\: x^{2} > \frac{3}{2} \\\frac{\sqrt{6} \operatorname{atanh}{\left(\frac{\sqrt{6} x}{3} \right)}}{6} & \text{for}\: x^{2} < \frac{3}{2} \end{cases}$$
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 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
12 12 12 12
$$- \frac{\sqrt{6} \log{\left(\frac{\sqrt{6}}{2} \right)}}{12} + \frac{\sqrt{6} \log{\left(1 + \frac{\sqrt{6}}{2} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(-1 + \frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(\frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12}$$
=
=
/ / ___\\ / ___\ / / ___\\ / ___\
___ | | \/ 6 || ___ |\/ 6 | ___ | |\/ 6 || ___ | \/ 6 |
\/ 6 *|pi*I + log|-1 + -----|| \/ 6 *log|-----| \/ 6 *|pi*I + log|-----|| \/ 6 *log|1 + -----|
\ \ 2 // \ 2 / \ \ 2 // \ 2 /
- ------------------------------ - ---------------- + ------------------------- + --------------------
12 12 12 12
$$- \frac{\sqrt{6} \log{\left(\frac{\sqrt{6}}{2} \right)}}{12} + \frac{\sqrt{6} \log{\left(1 + \frac{\sqrt{6}}{2} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(-1 + \frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(\frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12}$$
-sqrt(6)*(pi*i + log(-1 + sqrt(6)/2))/12 - sqrt(6)*log(sqrt(6)/2)/12 + sqrt(6)*(pi*i + log(sqrt(6)/2))/12 + sqrt(6)*log(1 + sqrt(6)/2)/12
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.