簡約 1/3 (ln(x+1)-ln(sqrt(x^2-x+1)))

解

簡約

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

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

解答ステップ

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

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

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

簡素化

\frac{x + 1}{x ^{2} - x + 1} : \frac{( x + 1 ) x ^{2} - x + 1}{x ^{2} - x + 1}

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

分数を乗じる:

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

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

乗算:

1 \cdot ln (\frac{( x + 1 ) x ^{2} - x + 1}{x ^{2} - x + 1}) = ln (\frac{( x + 1 ) x ^{2} - x + 1}{x ^{2} - x + 1})

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

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

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

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

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