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