integral of (2x-3)(5x^2-1)

解

\int (2 x - 3) (5 x^{2} - 1) d x

\frac{5 x ^{4}}{2} - x^{2} - 5 x^{3} + 3 x + C

解答ステップ

\int (2 x - 3) (5 x^{2} - 1) d x

拡張

(2 x - 3) (5 x^{2} - 1) : 10 x^{3} - 2 x - 15 x^{2} + 3

= \int 10 x^{3} - 2 x - 15 x^{2} + 3 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 10 x^{3} d x - \int 2 x d x - \int 15 x^{2} d x + \int 3 d x

\int 10 x^{3} d x = \frac{5 x ^{4}}{2}

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

\int 15 x^{2} d x = 5 x^{3}

\int 3 d x = 3 x

= \frac{5 x ^{4}}{2} - x^{2} - 5 x^{3} + 3 x

定数を解答に追加する

= \frac{5 x ^{4}}{2} - x^{2} - 5 x^{3} + 3 x + C