integral of (1-x^8)/(x(1+x^8))

解

\int \frac{1 - x ^{8}}{x ( 1 + x ^{8} )} d x

ln (x) - \frac{1}{4} ln 1 + x^{8} + C

解答ステップ

\int \frac{1 - x ^{8}}{x ( 1 + x ^{8} )} d x

拡張

\frac{1 - x ^{8}}{x ( 1 + x ^{8} )} : \frac{1}{x ( 1 + x ^{8} )} - \frac{x ^{8}}{x ( 1 + x ^{8} )}

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x ( 1 + x ^{8} )} d x - \int \frac{x ^{8}}{x ( 1 + x ^{8} )} d x

\int \frac{1}{x ( 1 + x ^{8} )} d x = \frac{1}{8} (8 ln (x) - ln 1 + x^{8})

\int \frac{x ^{8}}{x ( 1 + x ^{8} )} d x = \frac{1}{8} ln 1 + x^{8}

= \frac{1}{8} (8 ln (x) - ln 1 + x^{8}) - \frac{1}{8} ln 1 + x^{8}

簡素化

\frac{1}{8} (8 ln (x) - ln 1 + x^{8}) - \frac{1}{8} ln 1 + x^{8} : ln (x) - \frac{1}{4} ln 1 + x^{8}

= ln (x) - \frac{1}{4} ln 1 + x^{8}

定数を解答に追加する

= ln (x) - \frac{1}{4} ln 1 + x^{8} + C