Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sqrt(x^2-a^2)/2+x^2/(2*sqrt(x^2-a^2))-a^2*(1+x/sqrt(x^2-a^2))/(2*(x+sqrt(x^2-a^2)))
¿Cómo vas a descomponer esta sqrt(x^2-a^2)/2+x^2/(2*sqrt(x^2-a^2))-a^2*(1+x/sqrt(x^2-a^2))/(2*(x+sqrt(x^2-a^2))) expresión en fracciones?
Solución
Ha introducido
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2 / x \
a *|1 + ------------|
_________ | _________|
/ 2 2 2 | / 2 2 |
\/ x - a x \ \/ x - a /
------------ + -------------- - ---------------------
2 _________ / _________\
/ 2 2 | / 2 2 |
2*\/ x - a 2*\x + \/ x - a /
$$- \frac{a^{2} \left(\frac{x}{\sqrt{- a^{2} + x^{2}}} + 1\right)}{2 \left(x + \sqrt{- a^{2} + x^{2}}\right)} + \left(\frac{x^{2}}{2 \sqrt{- a^{2} + x^{2}}} + \frac{\sqrt{- a^{2} + x^{2}}}{2}\right)$$
sqrt(x^2 - a^2)/2 + x^2/((2*sqrt(x^2 - a^2))) - a^2*(1 + x/sqrt(x^2 - a^2))/(2*(x + sqrt(x^2 - a^2)))
Respuesta numérica
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0.5*(x^2 - a^2)^0.5 + 0.5*x^2*(x^2 - a^2)^(-0.5) - a^2*(1.0 + x*(x^2 - a^2)^(-0.5))/(2.0*x + 2.0*(x^2 - a^2)^0.5)
0.5*(x^2 - a^2)^0.5 + 0.5*x^2*(x^2 - a^2)^(-0.5) - a^2*(1.0 + x*(x^2 - a^2)^(-0.5))/(2.0*x + 2.0*(x^2 - a^2)^0.5)
Denominador racional
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3/2 _________ 3/2 _________
2 / 2 2\ 4 / 2 2 2 / 2 2\ 2 2 / 2 2
- 16*a *\x - a / - 16*x *\/ x - a + 16*x *\x - a / + 16*a *x *\/ x - a
--------------------------------------------------------------------------------------
2 / 2 2\
16*a *\a - x /
$$\frac{16 a^{2} x^{2} \sqrt{- a^{2} + x^{2}} - 16 a^{2} \left(- a^{2} + x^{2}\right)^{\frac{3}{2}} - 16 x^{4} \sqrt{- a^{2} + x^{2}} + 16 x^{2} \left(- a^{2} + x^{2}\right)^{\frac{3}{2}}}{16 a^{2} \left(a^{2} - x^{2}\right)}$$
(-16*a^2*(x^2 - a^2)^(3/2) - 16*x^4*sqrt(x^2 - a^2) + 16*x^2*(x^2 - a^2)^(3/2) + 16*a^2*x^2*sqrt(x^2 - a^2))/(16*a^2*(a^2 - x^2))
Combinatoria
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_________________ \/ (a + x)*(x - a)
$$\sqrt{\left(- a + x\right) \left(a + x\right)}$$
sqrt((a + x)*(x - a))
Unión de expresiones racionales
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_________ / 2 2 \/ x - a
$$\sqrt{- a^{2} + x^{2}}$$
sqrt(x^2 - a^2)
Potencias
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2 / x \
a *|-1 - ------------|
_________ | _________|
/ 2 2 2 | / 2 2 |
\/ x - a x \ \/ x - a /
------------ + -------------- + ----------------------
2 _________ _________
/ 2 2 / 2 2
2*\/ x - a 2*x + 2*\/ x - a
$$\frac{a^{2} \left(- \frac{x}{\sqrt{- a^{2} + x^{2}}} - 1\right)}{2 x + 2 \sqrt{- a^{2} + x^{2}}} + \frac{x^{2}}{2 \sqrt{- a^{2} + x^{2}}} + \frac{\sqrt{- a^{2} + x^{2}}}{2}$$
2 / x \
a *|1 + ------------|
_________ | _________|
/ 2 2 2 | / 2 2 |
\/ x - a x \ \/ x - a /
------------ + -------------- - ---------------------
2 _________ _________
/ 2 2 / 2 2
2*\/ x - a 2*x + 2*\/ x - a
$$- \frac{a^{2} \left(\frac{x}{\sqrt{- a^{2} + x^{2}}} + 1\right)}{2 x + 2 \sqrt{- a^{2} + x^{2}}} + \frac{x^{2}}{2 \sqrt{- a^{2} + x^{2}}} + \frac{\sqrt{- a^{2} + x^{2}}}{2}$$
sqrt(x^2 - a^2)/2 + x^2/(2*sqrt(x^2 - a^2)) - a^2*(1 + x/sqrt(x^2 - a^2))/(2*x + 2*sqrt(x^2 - a^2))
Denominador común
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2
a
x - ----------------
_________
/ 2 2
x + \/ x - a
$$- \frac{a^{2}}{x + \sqrt{- a^{2} + x^{2}}} + x$$
x - a^2/(x + sqrt(x^2 - a^2))
Parte trigonométrica
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2 / x \
a *|1 + ------------|
_________ | _________|
/ 2 2 2 | / 2 2 |
\/ x - a x \ \/ x - a /
------------ + -------------- - ---------------------
2 _________ _________
/ 2 2 / 2 2
2*\/ x - a 2*x + 2*\/ x - a
$$- \frac{a^{2} \left(\frac{x}{\sqrt{- a^{2} + x^{2}}} + 1\right)}{2 x + 2 \sqrt{- a^{2} + x^{2}}} + \frac{x^{2}}{2 \sqrt{- a^{2} + x^{2}}} + \frac{\sqrt{- a^{2} + x^{2}}}{2}$$
sqrt(x^2 - a^2)/2 + x^2/(2*sqrt(x^2 - a^2)) - a^2*(1 + x/sqrt(x^2 - a^2))/(2*x + 2*sqrt(x^2 - a^2))
Compilar la expresión
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2 / x \
a *|1 + ------------|
_________ | _________|
/ 2 2 2 | / 2 2 |
\/ x - a x \ \/ x - a /
------------ + -------------- - ---------------------
2 _________ _________
/ 2 2 / 2 2
2*\/ x - a 2*x + 2*\/ x - a
$$- \frac{a^{2} \left(\frac{x}{\sqrt{- a^{2} + x^{2}}} + 1\right)}{2 x + 2 \sqrt{- a^{2} + x^{2}}} + \frac{x^{2}}{2 \sqrt{- a^{2} + x^{2}}} + \frac{\sqrt{- a^{2} + x^{2}}}{2}$$
sqrt(x^2 - a^2)/2 + x^2/(2*sqrt(x^2 - a^2)) - a^2*(1 + x/sqrt(x^2 - a^2))/(2*x + 2*sqrt(x^2 - a^2))