Sr Examen
Integral de ln(t)/(1-t^2) dt
Solución
Ha introducido
[src]
1/5 / | | log(t) | ------ dt | 2 | 1 - t | / 1/10
$$\int\limits_{\frac{1}{10}}^{\frac{1}{5}} \frac{\log{\left(t \right)}}{1 - t^{2}}\, dt$$
Integral(log(t)/(1 - t^2), (t, 1/10, 1/5))
Respuesta (Indefinida)
[src]
// / \
|| | |
|| | acoth(t) |
|| - | -------- dt for t < -1|
|| | t |
|| | |
|| / |
|| |
|| -1 -1 |
|| / / / |
/ || | | | |
| // 2 \ || | acoth(t) | atanh(t) | atanh(t) |
| log(t) ||-acoth(t) for t > 1| || - | -------- dt - | -------- dt + | -------- dt for t < 1 |
| ------ dt = C - |< |*log(t) + |< | t | t | t |
| 2 || 2 | || | | | |
| 1 - t \\-atanh(t) for t < 1/ || / / / |
| || |
/ || |
|| -1 1 1 -1 |
|| / / / / / |
|| | | | | | |
|| | acoth(t) | acoth(t) | atanh(t) | acoth(t) | atanh(t) |
||- | -------- dt - | -------- dt - | -------- dt + | -------- dt + | -------- dt otherwise |
|| | t | t | t | t | t |
|| | | | | | |
|| / / / / / |
\\ /
$$\int \frac{\log{\left(t \right)}}{1 - t^{2}}\, dt = C - \left(\begin{cases} - \operatorname{acoth}{\left(t \right)} & \text{for}\: t^{2} > 1 \\- \operatorname{atanh}{\left(t \right)} & \text{for}\: t^{2} < 1 \end{cases}\right) \log{\left(t \right)} + \begin{cases} - \int \frac{\operatorname{acoth}{\left(t \right)}}{t}\, dt & \text{for}\: t < -1 \\- \int\limits^{-1} \frac{\operatorname{acoth}{\left(t \right)}}{t}\, dt - \int \frac{\operatorname{atanh}{\left(t \right)}}{t}\, dt + \int\limits^{-1} \frac{\operatorname{atanh}{\left(t \right)}}{t}\, dt & \text{for}\: t < 1 \\- \int \frac{\operatorname{acoth}{\left(t \right)}}{t}\, dt - \int\limits^{-1} \frac{\operatorname{acoth}{\left(t \right)}}{t}\, dt + \int\limits^{1} \frac{\operatorname{acoth}{\left(t \right)}}{t}\, dt + \int\limits^{-1} \frac{\operatorname{atanh}{\left(t \right)}}{t}\, dt - \int\limits^{1} \frac{\operatorname{atanh}{\left(t \right)}}{t}\, dt & \text{otherwise} \end{cases}$$
Respuesta
[src]
1/5
/
|
| log(t)
- | ------- dt
| 2
| -1 + t
|
/
1/10
$$- \int\limits_{\frac{1}{10}}^{\frac{1}{5}} \frac{\log{\left(t \right)}}{t^{2} - 1}\, dt$$
=
=
1/5
/
|
| log(t)
- | ------- dt
| 2
| -1 + t
|
/
1/10
$$- \int\limits_{\frac{1}{10}}^{\frac{1}{5}} \frac{\log{\left(t \right)}}{t^{2} - 1}\, dt$$
-Integral(log(t)/(-1 + t^2), (t, 1/10, 1/5))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.