Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
log((sqrt(1-a^2/x^2)-1)/(sqrt(1-a^2/x^2)+1))
¿Cómo vas a descomponer esta log((sqrt(1-a^2/x^2)-1)/(sqrt(1-a^2/x^2)+1)) expresión en fracciones?
Solución
Ha introducido
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/ ________ \ | / 2 | | / a | | / 1 - -- - 1| | / 2 | |\/ x | log|------------------| | ________ | | / 2 | | / a | | / 1 - -- + 1| | / 2 | \\/ x /
$$\log{\left(\frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1} - 1}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} \right)}$$
log((sqrt(1 - a^2/x^2) - 1)/(sqrt(1 - a^2/x^2) + 1))
Descomposición de una fracción
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log(-1/(1 + sqrt(1 - a^2/x^2)) + sqrt(1 - a^2/x^2)/(1 + sqrt(1 - a^2/x^2)))
$$\log{\left(\frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} - \frac{1}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} \right)}$$
/ ________ \ | / 2 | | / a | | / 1 - -- | | / 2 | | 1 \/ x | log|- ------------------ + ------------------| | ________ ________| | / 2 / 2 | | / a / a | | 1 + / 1 - -- 1 + / 1 - -- | | / 2 / 2 | \ \/ x \/ x /
Respuesta numérica
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log((sqrt(1 - a^2/x^2) - 1)/(sqrt(1 - a^2/x^2) + 1))
log((sqrt(1 - a^2/x^2) - 1)/(sqrt(1 - a^2/x^2) + 1))
Denominador común
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/ ________ \ | / 2 | | / a | | / 1 - -- | | / 2 | | 1 \/ x | log|- ------------------ + ------------------| | ________ ________| | / 2 / 2 | | / a / a | | 1 + / 1 - -- 1 + / 1 - -- | | / 2 / 2 | \ \/ x \/ x /
$$\log{\left(\frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} - \frac{1}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} \right)}$$
log(-1/(1 + sqrt(1 - a^2/x^2)) + sqrt(1 - a^2/x^2)/(1 + sqrt(1 - a^2/x^2)))
Unión de expresiones racionales
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/ _________\ | / 2 2 | | / x - a | |-1 + / ------- | | / 2 | | \/ x | log|--------------------| | _________ | | / 2 2 | | / x - a | |1 + / ------- | | / 2 | \ \/ x /
$$\log{\left(\frac{\sqrt{\frac{- a^{2} + x^{2}}{x^{2}}} - 1}{\sqrt{\frac{- a^{2} + x^{2}}{x^{2}}} + 1} \right)}$$
log((-1 + sqrt((x^2 - a^2)/x^2))/(1 + sqrt((x^2 - a^2)/x^2)))
Combinatoria
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/ ________ \ | / 2 | | / a | | / 1 - -- | | / 2 | | 1 \/ x | log|- ------------------ + ------------------| | ________ ________| | / 2 / 2 | | / a / a | | 1 + / 1 - -- 1 + / 1 - -- | | / 2 / 2 | \ \/ x \/ x /
$$\log{\left(\frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} - \frac{1}{\sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1} \right)}$$
log(-1/(1 + sqrt(1 - a^2/x^2)) + sqrt(1 - a^2/x^2)/(1 + sqrt(1 - a^2/x^2)))
Denominador racional
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/ / ________ ________ ________ ________\ \ | | / 2 / 2 / 2 / 2 | | | 2 | / a / a / a / a | | |-x *|1 - / 1 - -- - / 1 - -- + / 1 - -- * / 1 - -- | | | | / 2 / 2 / 2 / 2 | | | \ \/ x \/ x \/ x \/ x / | log|--------------------------------------------------------------------------| | 2 | \ a /
$$\log{\left(- \frac{x^{2} \left(\sqrt{- \frac{a^{2}}{x^{2}} + 1} \sqrt{- \frac{a^{2}}{x^{2}} + 1} - \sqrt{- \frac{a^{2}}{x^{2}} + 1} - \sqrt{- \frac{a^{2}}{x^{2}} + 1} + 1\right)}{a^{2}} \right)}$$
log(-x^2*(1 - sqrt(1 - a^2/x^2) - sqrt(1 - a^2/x^2) + sqrt(1 - a^2/x^2)*sqrt(1 - a^2/x^2))/a^2)