Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
Piecewise((-a^2*acosh(x/a)/2+x^3/(2*a*sqrt(-1+x^2/a^2))-a*x/(2*sqrt(-1+x^2/a^2)),|x^2/a^2|>1),(i*a^2*asin(x/a)/2+i*a*x*sqrt(1-x^2/a^2)/2,True))
¿Cómo vas a descomponer esta Piecewise((-a^2*acosh(x/a)/2+x^3/(2*a*sqrt(-1+x^2/a^2))-a*x/(2*sqrt(-1+x^2/a^2)),|x^2/a^2|>1),(i*a^2*asin(x/a)/2+i*a*x*sqrt(1-x^2/a^2)/2,True)) expresión en fracciones?
Solución
Ha introducido
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/ 2 /x\ | a *acosh|-| 3 | 2| | \a/ x a*x |x | |- ----------- + ------------------- - ----------------- for |--| > 1 | 2 _________ _________ | 2| | / 2 / 2 |a | | / x / x | 2*a* / -1 + -- 2* / -1 + -- | / 2 / 2 < \/ a \/ a | | ________ | / 2 | / x | 2 /x\ I*a*x* / 1 - -- | I*a *asin|-| / 2 | \a/ \/ a | ------------ + -------------------- otherwise \ 2 2
$$\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{2} - \frac{a x}{2 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{x^{3}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} + \frac{i a x \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}$$
Piecewise((-a^2*acosh(x/a)/2 + x^3/(2*a*sqrt(-1 + x^2/a^2)) - a*x/(2*sqrt(-1 + x^2/a^2)), |x^2/a^2| > 1), (i*a^2*asin(x/a)/2 + i*a*x*sqrt(1 - x^2/a^2)/2, True))
Simplificación general
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/ / _________ \ | | / 2 2 | | | / x - a /x\| |a*|x* / ------- - a*acosh|-|| | | / 2 \a/| | 2| | \ \/ a / |x | |---------------------------------- for |--| > 1 | 2 | 2| | |a | < | / _________\ | | / 2 2 | | | /x\ / a - x | |I*a*|a*asin|-| + x* / ------- | | | \a/ / 2 | | \ \/ a / |----------------------------------- otherwise | 2 \
$$\begin{cases} \frac{a \left(- a \operatorname{acosh}{\left(\frac{x}{a} \right)} + x \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}}\right)}{2} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{i a \left(a \operatorname{asin}{\left(\frac{x}{a} \right)} + x \sqrt{\frac{a^{2} - x^{2}}{a^{2}}}\right)}{2} & \text{otherwise} \end{cases}$$
Piecewise((a*(x*sqrt((x^2 - a^2)/a^2) - a*acosh(x/a))/2, |x^2/a^2| > 1), (i*a*(a*asin(x/a) + x*sqrt((a^2 - x^2)/a^2))/2, True))
Respuesta numérica
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Piecewise((-0.5*a^2*acosh(x/a) + 0.5*x^3*(-1.0 + x^2/a^2)^(-0.5)/a - 0.5*a*x*(-1.0 + x^2/a^2)^(-0.5), |x^2/a^2| > 1), (0.5*i*a^2*asin(x/a) + 0.5*i*a*x*(1.0 - x^2/a^2)^0.5, True))
Piecewise((-0.5*a^2*acosh(x/a) + 0.5*x^3*(-1.0 + x^2/a^2)^(-0.5)/a - 0.5*a*x*(-1.0 + x^2/a^2)^(-0.5), |x^2/a^2| > 1), (0.5*i*a^2*asin(x/a) + 0.5*i*a*x*(1.0 - x^2/a^2)^0.5, True))
Unión de expresiones racionales
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/ _________ | / 2 2 | 3 2 3 / x - a /x\ |x - x*a - a * / ------- *acosh|-| | / 2 \a/ | 2| | \/ a |x | |--------------------------------------- for |--| > 1 | _________ | 2| | / 2 2 |a | | / x - a | 2*a* / ------- < / 2 | \/ a | | / _________\ | | / 2 2 | | | /x\ / a - x | | I*a*|a*asin|-| + x* / ------- | | | \a/ / 2 | | \ \/ a / | ----------------------------------- otherwise | 2 \
$$\begin{cases} \frac{- a^{3} \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}} \operatorname{acosh}{\left(\frac{x}{a} \right)} - a^{2} x + x^{3}}{2 a \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{i a \left(a \operatorname{asin}{\left(\frac{x}{a} \right)} + x \sqrt{\frac{a^{2} - x^{2}}{a^{2}}}\right)}{2} & \text{otherwise} \end{cases}$$
Piecewise(((x^3 - x*a^2 - a^3*sqrt((x^2 - a^2)/a^2)*acosh(x/a))/(2*a*sqrt((x^2 - a^2)/a^2)), |x^2/a^2| > 1), (i*a*(a*asin(x/a) + x*sqrt((a^2 - x^2)/a^2))/2, True))
Potencias
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/ 2 /x\ | a *acosh|-| 3 | 2| | \a/ x a*x |x | |- ----------- + ------------------- - ----------------- for |--| > 1 | 2 _________ _________ | 2| | / 2 / 2 |a | | / x / x | 2*a* / -1 + -- 2* / -1 + -- | / 2 / 2 < \/ a \/ a | | _________ | / 2 | / x | 2 /x\ a*x* / -1 + -- | I*a *asin|-| / 2 | \a/ \/ a | ------------ + ------------------- otherwise \ 2 2
$$\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{2} - \frac{a x}{2 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{x^{3}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{i a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} + \frac{a x \sqrt{-1 + \frac{x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}$$
Piecewise((-a^2*acosh(x/a)/2 + x^3/(2*a*sqrt(-1 + x^2/a^2)) - a*x/(2*sqrt(-1 + x^2/a^2)), |x^2/a^2| > 1), (i*a^2*asin(x/a)/2 + a*x*sqrt(-1 + x^2/a^2)/2, True))