integral of (x^5)/(1-x^4)

解

\int \frac{x ^{5}}{1 - x ^{4}} d x

- \frac{x ^{2}}{2} + \frac{1}{4} (ln x^{2} + 1 - ln x^{2} - 1) + C

解答ステップ

\int \frac{x ^{5}}{1 - x ^{4}} d x

長除法

\frac{x ^{5}}{1 - x ^{4}} : - x + \frac{x}{- x ^{4} + 1}

= \int - x + \frac{x}{- x ^{4} + 1} d x

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 x d x + \int \frac{x}{- x ^{4} + 1} d x

\int x d x = \frac{x ^{2}}{2}

\int \frac{x}{- x ^{4} + 1} d x = \frac{1}{4} (ln x^{2} + 1 - ln x^{2} - 1)

= - \frac{x ^{2}}{2} + \frac{1}{4} (ln x^{2} + 1 - ln x^{2} - 1)

定数を解答に追加する

= - \frac{x ^{2}}{2} + \frac{1}{4} (ln x^{2} + 1 - ln x^{2} - 1) + C