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