Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
Piecewise((-acosh(1/x)+1/(x*sqrt(-1+x^(-2)))-x/sqrt(-1+x^(-2)),1/|x^2|>1),(i*asin(1/x)+i*x/sqrt(1-1/x^2)-i/(x*sqrt(1-1/x^2)),True))
¿Cómo vas a descomponer esta Piecewise((-acosh(1/x)+1/(x*sqrt(-1+x^(-2)))-x/sqrt(-1+x^(-2)),1/|x^2|>1),(i*asin(1/x)+i*x/sqrt(1-1/x^2)-i/(x*sqrt(1-1/x^2)),True)) expresión en fracciones?
Solución
Ha introducido
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/ /1\ 1 x 1 |- acosh|-| + ---------------- - -------------- for ---- > 1 | \x/ _________ _________ | 2| | / 1 / 1 |x | | x* / -1 + -- / -1 + -- | / 2 / 2 | \/ x \/ x < | /1\ I*x I | I*asin|-| + ------------- - --------------- otherwise | \x/ ________ ________ | / 1 / 1 | / 1 - -- x* / 1 - -- | / 2 / 2 \ \/ x \/ x
$$\begin{cases} - \frac{x}{\sqrt{-1 + \frac{1}{x^{2}}}} - \operatorname{acosh}{\left(\frac{1}{x} \right)} + \frac{1}{x \sqrt{-1 + \frac{1}{x^{2}}}} & \text{for}\: \frac{1}{\left|{x^{2}}\right|} > 1 \\\frac{i x}{\sqrt{1 - \frac{1}{x^{2}}}} + i \operatorname{asin}{\left(\frac{1}{x} \right)} - \frac{i}{x \sqrt{1 - \frac{1}{x^{2}}}} & \text{otherwise} \end{cases}$$
Piecewise((-acosh(1/x) + 1/(x*sqrt(-1 + x^(-2))) - x/sqrt(-1 + x^(-2)), 1/|x^2| > 1), (i*asin(1/x) + i*x/sqrt(1 - 1/x^2) - i/(x*sqrt(1 - 1/x^2)), True))
Simplificación general
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/ _________ | /1\ / 1 1 | - acosh|-| + x* / -1 + -- for ---- > 1 | \x/ / 2 | 2| | \/ x |x | | < / _________ \ | | / 2 | | | / -1 + x /1\| |I*|x* / ------- + asin|-|| otherwise | | / 2 \x/| | \ \/ x / \
$$\begin{cases} x \sqrt{-1 + \frac{1}{x^{2}}} - \operatorname{acosh}{\left(\frac{1}{x} \right)} & \text{for}\: \frac{1}{\left|{x^{2}}\right|} > 1 \\i \left(x \sqrt{\frac{x^{2} - 1}{x^{2}}} + \operatorname{asin}{\left(\frac{1}{x} \right)}\right) & \text{otherwise} \end{cases}$$
Piecewise((-acosh(1/x) + x*sqrt(-1 + x^(-2)), 1/|x^2| > 1), (i*(x*sqrt((-1 + x^2)/x^2) + asin(1/x)), True))
Unión de expresiones racionales
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/ ________ | / 2 | 2 / 1 - x /1\ | 1 - x - x* / ------ *acosh|-| | / 2 \x/ | \/ x 1 | ---------------------------------- for ---- > 1 | ________ | 2| | / 2 |x | | / 1 - x | x* / ------ | / 2 | \/ x < | / _________ \ | | / 2 | | | 2 / -1 + x /1\| |I*|-1 + x + x* / ------- *asin|-|| | | / 2 \x/| | \ \/ x / |--------------------------------------- otherwise | _________ | / 2 | / -1 + x | x* / ------- | / 2 \ \/ x
$$\begin{cases} \frac{- x^{2} - x \sqrt{\frac{1 - x^{2}}{x^{2}}} \operatorname{acosh}{\left(\frac{1}{x} \right)} + 1}{x \sqrt{\frac{1 - x^{2}}{x^{2}}}} & \text{for}\: \frac{1}{\left|{x^{2}}\right|} > 1 \\\frac{i \left(x^{2} + x \sqrt{\frac{x^{2} - 1}{x^{2}}} \operatorname{asin}{\left(\frac{1}{x} \right)} - 1\right)}{x \sqrt{\frac{x^{2} - 1}{x^{2}}}} & \text{otherwise} \end{cases}$$
Piecewise(((1 - x^2 - x*sqrt((1 - x^2)/x^2)*acosh(1/x))/(x*sqrt((1 - x^2)/x^2)), 1/|x^2| > 1), (i*(-1 + x^2 + x*sqrt((-1 + x^2)/x^2)*asin(1/x))/(x*sqrt((-1 + x^2)/x^2)), True))
Respuesta numérica
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Piecewise((-acosh(1/x) + (-1.0 + x^(-2))^(-0.5)/x - x*(-1.0 + x^(-2))^(-0.5), 1/|x^2| > 1), (i*asin(1/x) + i*x*(1.0 - 1/x^2)^(-0.5) - i*(1.0 - 1/x^2)^(-0.5)/x, True))
Piecewise((-acosh(1/x) + (-1.0 + x^(-2))^(-0.5)/x - x*(-1.0 + x^(-2))^(-0.5), 1/|x^2| > 1), (i*asin(1/x) + i*x*(1.0 - 1/x^2)^(-0.5) - i*(1.0 - 1/x^2)^(-0.5)/x, True))