Sr Examen
Integral de 1/(3x^2-9) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------- dx | 2 | 3*x - 9 | / 0
$$\int\limits_{0}^{1} \frac{1}{3 x^{2} - 9}\, dx$$
Integral(1/(3*x^2 - 9), (x, 0, 1))
Solución detallada
-
Añadimos la constante de integración:
PieceweseRule(subfunctions=[(ArctanRule(a=1, b=3, c=-9, context=1/(3*x**2 - 9), symbol=x), False), (ArccothRule(a=1, b=3, c=-9, context=1/(3*x**2 - 9), symbol=x), x**2 > 3), (ArctanhRule(a=1, b=3, c=-9, context=1/(3*x**2 - 9), symbol=x), x**2 < 3)], context=1/(3*x**2 - 9), symbol=x)
Respuesta:
Respuesta (Indefinida)
[src]
// / ___\ \
|| ___ |x*\/ 3 | |
||-\/ 3 *acoth|-------| |
/ || \ 3 / 2 |
| ||---------------------- for x > 3|
| 1 || 9 |
| -------- dx = C + |< |
| 2 || / ___\ |
| 3*x - 9 || ___ |x*\/ 3 | |
| ||-\/ 3 *atanh|-------| |
/ || \ 3 / 2 |
||---------------------- for x < 3|
\\ 9 /
$$\int \frac{1}{3 x^{2} - 9}\, dx = C + \begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} x}{3} \right)}}{9} & \text{for}\: x^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} x}{3} \right)}}{9} & \text{for}\: x^{2} < 3 \end{cases}$$
Gráfica
Respuesta
[src]
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\
\/ 3 *\pi*I + log\\/ 3 // \/ 3 *log\1 + \/ 3 / \/ 3 *\pi*I + log\-1 + \/ 3 // \/ 3 *log\\/ 3 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
18 18 18 18
$$- \frac{\sqrt{3} \log{\left(1 + \sqrt{3} \right)}}{18} + \frac{\sqrt{3} \log{\left(\sqrt{3} \right)}}{18} - \frac{\sqrt{3} \left(\log{\left(\sqrt{3} \right)} + i \pi\right)}{18} + \frac{\sqrt{3} \left(\log{\left(-1 + \sqrt{3} \right)} + i \pi\right)}{18}$$
=
=
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\
\/ 3 *\pi*I + log\\/ 3 // \/ 3 *log\1 + \/ 3 / \/ 3 *\pi*I + log\-1 + \/ 3 // \/ 3 *log\\/ 3 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
18 18 18 18
$$- \frac{\sqrt{3} \log{\left(1 + \sqrt{3} \right)}}{18} + \frac{\sqrt{3} \log{\left(\sqrt{3} \right)}}{18} - \frac{\sqrt{3} \left(\log{\left(\sqrt{3} \right)} + i \pi\right)}{18} + \frac{\sqrt{3} \left(\log{\left(-1 + \sqrt{3} \right)} + i \pi\right)}{18}$$
-sqrt(3)*(pi*i + log(sqrt(3)))/18 - sqrt(3)*log(1 + sqrt(3))/18 + sqrt(3)*(pi*i + log(-1 + sqrt(3)))/18 + sqrt(3)*log(sqrt(3))/18
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.