簡約 1/(4(1+x^4))+ln(x)-1/4 ln(1+x^4)

解

簡約

\frac{1}{4 ( 1 + x ^{4} )} + ln (x) - \frac{1}{4} ln (1 + x^{4})

\frac{1}{4 ( 1 + x ^{4} )} + ln \frac{x ( 1 + x ^{4} ) ^{\frac{3}{4}}}{1 + x ^{4}}

解答ステップ

\frac{1}{4 ( 1 + x ^{4} )} + ln (x) - \frac{1}{4} ln (1 + x^{4})

対数の規則を適用する:

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

\frac{1}{4} ln (x^{4} + 1) = ln ((x^{4} + 1)^{\frac{1}{4}})

= \frac{1}{4 ( x ^{4} + 1 )} + ln (x) - ln ((x^{4} + 1)^{\frac{1}{4}})

対数の規則を適用する:

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

ln (x) - ln ((1 + x^{4})^{\frac{1}{4}}) = ln (\frac{x}{( x ^{4} + 1 ) ^{\frac{1}{4}}})

= \frac{1}{4 ( x ^{4} + 1 )} + ln (\frac{x}{( x ^{4} + 1 ) ^{\frac{1}{4}}})

簡素化

\frac{x}{( 1 + x ^{4} ) ^{\frac{1}{4}}} : \frac{x ( 1 + x ^{4} ) ^{\frac{3}{4}}}{1 + x ^{4}}

= \frac{1}{4 ( 1 + x ^{4} )} + ln \frac{x ( 1 + x ^{4} ) ^{\frac{3}{4}}}{1 + x ^{4}}