Sr Examen
Integral de 1/(sqrt(y^2/c-1)) dy
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dy | ________ | / 2 | / y | / -- - 1 | \/ c | / 0
$$\int\limits_{0}^{1} \frac{1}{\sqrt{-1 + \frac{y^{2}}{c}}}\, dy$$
Integral(1/(sqrt(y^2/c - 1)), (y, 0, 1))
Respuesta (Indefinida)
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// | 2| \
|| ___ / y \ |y | |
/ || \/ c *acosh|-----| for |--| > 1|
| || | ___| |c | |
| 1 || \\/ c / |
| ------------- dy = C + |< |
| ________ || ___ / y \ |
| / 2 ||-I*\/ c *asin|-----| otherwise |
| / y || | ___| |
| / -- - 1 || \\/ c / |
| \/ c \\ /
|
/
$$\int \frac{1}{\sqrt{-1 + \frac{y^{2}}{c}}}\, dy = C + \begin{cases} \sqrt{c} \operatorname{acosh}{\left(\frac{y}{\sqrt{c}} \right)} & \text{for}\: \left|{\frac{y^{2}}{c}}\right| > 1 \\- i \sqrt{c} \operatorname{asin}{\left(\frac{y}{\sqrt{c}} \right)} & \text{otherwise} \end{cases}$$
Respuesta
[src]
1 / | | / 2 | | 1 y | |-------------- for --- > 1 | | _________ |c| | | / 2 | | / y | | / -1 + -- | |\/ c | < dy | | -I | |------------- otherwise | | ________ | | / 2 | | / y | | / 1 - -- | |\/ c | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{1}{\sqrt{-1 + \frac{y^{2}}{c}}} & \text{for}\: \frac{y^{2}}{\left|{c}\right|} > 1 \\- \frac{i}{\sqrt{1 - \frac{y^{2}}{c}}} & \text{otherwise} \end{cases}\, dy$$
=
=
1 / | | / 2 | | 1 y | |-------------- for --- > 1 | | _________ |c| | | / 2 | | / y | | / -1 + -- | |\/ c | < dy | | -I | |------------- otherwise | | ________ | | / 2 | | / y | | / 1 - -- | |\/ c | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{1}{\sqrt{-1 + \frac{y^{2}}{c}}} & \text{for}\: \frac{y^{2}}{\left|{c}\right|} > 1 \\- \frac{i}{\sqrt{1 - \frac{y^{2}}{c}}} & \text{otherwise} \end{cases}\, dy$$
Integral(Piecewise((1/sqrt(-1 + y^2/c), y^2/|c| > 1), (-i/sqrt(1 - y^2/c), True)), (y, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.