integral of (12x^{-1}-6x^4-9x^2+5x-15)

解

\int (12 x^{- 1} - 6 x^{4} - 9 x^{2} + 5 x - 15) d x

12 ln ∣ x ∣ - \frac{6 x ^{5}}{5} - 3 x^{3} + \frac{5 x ^{2}}{2} - 15 x + C

解答ステップ

\int 12 x^{- 1} - 6 x^{4} - 9 x^{2} + 5 x - 15 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 12 x^{- 1} d x - \int 6 x^{4} d x - \int 9 x^{2} d x + \int 5 x d x - \int 15 d x

\int 12 x^{- 1} d x = 12 ln ∣ x ∣

\int 6 x^{4} d x = \frac{6 x ^{5}}{5}

\int 9 x^{2} d x = 3 x^{3}

\int 5 x d x = \frac{5 x ^{2}}{2}

\int 15 d x = 15 x

= 12 ln ∣ x ∣ - \frac{6 x ^{5}}{5} - 3 x^{3} + \frac{5 x ^{2}}{2} - 15 x

定数を解答に追加する

= 12 ln ∣ x ∣ - \frac{6 x ^{5}}{5} - 3 x^{3} + \frac{5 x ^{2}}{2} - 15 x + C