Sr Examen
Integral de lnx/(x^2-1) dx
Solución
Ha introducido
[src]
1 / | | log(x) | ------ dx | 2 | x - 1 | / 0
$$\int\limits_{0}^{1} \frac{\log{\left(x \right)}}{x^{2} - 1}\, dx$$
Integral(log(x)/(x^2 - 1), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / \
|| | |
|| | acoth(x) |
|| - | -------- dx for x < -1|
|| | x |
|| | |
|| / |
|| |
|| -1 -1 |
|| / / / |
/ || | | | |
| || | acoth(x) | atanh(x) | atanh(x) | // 2 \
| log(x) || - | -------- dx - | -------- dx + | -------- dx for x < 1 | ||-acoth(x) for x > 1|
| ------ dx = C - |< | x | x | x | + |< |*log(x)
| 2 || | | | | || 2 |
| x - 1 || / / / | \\-atanh(x) for x < 1/
| || |
/ || |
|| -1 1 1 -1 |
|| / / / / / |
|| | | | | | |
|| | acoth(x) | acoth(x) | atanh(x) | acoth(x) | atanh(x) |
||- | -------- dx - | -------- dx - | -------- dx + | -------- dx + | -------- dx otherwise |
|| | x | x | x | x | x |
|| | | | | | |
|| / / / / / |
\\ /
$$\int \frac{\log{\left(x \right)}}{x^{2} - 1}\, dx = C + \left(\begin{cases} - \operatorname{acoth}{\left(x \right)} & \text{for}\: x^{2} > 1 \\- \operatorname{atanh}{\left(x \right)} & \text{for}\: x^{2} < 1 \end{cases}\right) \log{\left(x \right)} - \begin{cases} - \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx & \text{for}\: x < -1 \\- \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{for}\: x < 1 \\- \int \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx - \int\limits^{-1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{1} \frac{\operatorname{acoth}{\left(x \right)}}{x}\, dx + \int\limits^{-1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx - \int\limits^{1} \frac{\operatorname{atanh}{\left(x \right)}}{x}\, dx & \text{otherwise} \end{cases}$$
Respuesta
[src]
1 / | | log(x) | ---------------- dx | (1 + x)*(-1 + x) | / 0
$$\int\limits_{0}^{1} \frac{\log{\left(x \right)}}{\left(x - 1\right) \left(x + 1\right)}\, dx$$
=
=
1 / | | log(x) | ---------------- dx | (1 + x)*(-1 + x) | / 0
$$\int\limits_{0}^{1} \frac{\log{\left(x \right)}}{\left(x - 1\right) \left(x + 1\right)}\, dx$$
Integral(log(x)/((1 + x)*(-1 + x)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.