integral of u^{-5}(5-u^5+u^{10})

解

\int u^{- 5} (5 - u^{5} + u^{10}) d u

- \frac{5}{4 u ^{4}} - u + \frac{u ^{6}}{6} + C

解答ステップ

\int u^{- 5} (5 - u^{5} + u^{10}) d u

拡張

u^{- 5} (5 - u^{5} + u^{10}) : \frac{5}{u ^{5}} - 1 + u^{5}

= \int \frac{5}{u ^{5}} - 1 + u^{5} d u

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{5}{u ^{5}} d u - \int 1 d u + \int u^{5} d u

\int \frac{5}{u ^{5}} d u = - \frac{5}{4 u ^{4}}

\int 1 d u = u

\int u^{5} d u = \frac{u ^{6}}{6}

= - \frac{5}{4 u ^{4}} - u + \frac{u ^{6}}{6}

定数を解答に追加する

= - \frac{5}{4 u ^{4}} - u + \frac{u ^{6}}{6} + C