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simplificar e^{-2/3 (2ln|2x^3-1|-ln|x^3|)+C}

Solução

simplificar

e^{- \frac{2}{3} (2 l n ∣ 2 x^{3} - 1 ∣ - l n ∣ x^{3} ∣) + C}

\frac{e ^{C} 3 ( 2 x ^{3} - 1 ) ^{2} x ^{3} ^{\frac{2}{3}}}{4 x ^{6} - 4 x ^{3} + 1}

Passos da solução

e^{- \frac{2}{3} (2 l n ∣ 2 x^{3} - 1 ∣ - l n ∣ x^{3} ∣) + C}

Aplicar as propriedades dos logaritmos:

a lo g_{c} (b) = lo g_{c} (b^{a})

2 ln 2 x^{3} - 1 = ln (2 x^{3} - 1^{2})

= e^{- \frac{2}{3} (l n (∣ 2 x^{3} - 1 ∣^{2}) - l n ∣ x^{3} ∣) + C}

Aplicar as propriedades dos logaritmos:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (2 x^{3} - 1^{2}) - ln x^{3} = ln (\frac{2 x ^{3} - 1 ^{2}}{∣ x ^{3} ∣})

= e^{- \frac{2}{3} l n (\frac{∣ 2 x ^{3} - 1 ∣ ^{2}}{∣ x ^{3} ∣}) + C}

2 x^{3} - 1^{2} = (2 x^{3} - 1)^{2}

= e^{- \frac{2}{3} l n (\frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣}) + C}

Aplicar as propriedades dos expoentes:

a^{b + c} = a^{b} a^{c}

= e^{- \frac{2}{3} l n (\frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣})} e^{C}

Simplificar

e^{- \frac{2}{3} l n (\frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣})} : (\frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣})^{- \frac{2}{3}}

= (\frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣})^{- \frac{2}{3}} e^{C}

Aplicar as propriedades dos expoentes:

a^{- b} = \frac{1}{a ^{b}}

(\frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣})^{- \frac{2}{3}} = \frac{1}{( \frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣} ) ^{\frac{2}{3}}}

= e^{C} \frac{1}{( \frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣} ) ^{\frac{2}{3}}}

Multiplicar frações:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot e ^{C}}{( \frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣} ) ^{\frac{2}{3}}}

Multiplicar:

1 \cdot e^{C} = e^{C}

= \frac{e ^{C}}{( \frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣} ) ^{\frac{2}{3}}}

(\frac{( 2 x ^{3} - 1 ) ^{2}}{∣ x ^{3} ∣})^{\frac{2}{3}} = \frac{( ( 2 x ^{3} - 1 ) ^{2} ) ^{\frac{2}{3}}}{∣ x ^{3} ∣ ^{\frac{2}{3}}}

= \frac{e ^{C}}{\frac{( ( 2 x ^{3} - 1 ) ^{2} ) ^{\frac{2}{3}}}{∣ x ^{3} ∣ ^{\frac{2}{3}}}}

Aplicar as propriedades das frações:

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

= \frac{e ^{C} x ^{3} ^{\frac{2}{3}}}{( ( 2 x ^{3} - 1 ) ^{2} ) ^{\frac{2}{3}}}

Racionalizar

\frac{e ^{C} x ^{3} ^{\frac{2}{3}}}{( ( 2 x ^{3} - 1 ) ^{2} ) ^{\frac{2}{3}}} : \frac{e ^{C} 3 ( 2 x ^{3} - 1 ) ^{2} x ^{3} ^{\frac{2}{3}}}{4 x ^{6} - 4 x ^{3} + 1}

= \frac{e ^{C} 3 ( 2 x ^{3} - 1 ) ^{2} x ^{3} ^{\frac{2}{3}}}{4 x ^{6} - 4 x ^{3} + 1}

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