integral of (x+5)(x^3+5x-10)

解

\int (x + 5) (x^{3} + 5 x - 10) d x

\frac{x ^{5}}{5} + \frac{5 x ^{4}}{4} + \frac{5 x ^{3}}{3} + \frac{15 x ^{2}}{2} - 50 x + C

解答ステップ

\int (x + 5) (x^{3} + 5 x - 10) d x

拡張

(x + 5) (x^{3} + 5 x - 10) : x^{4} + 5 x^{3} + 5 x^{2} + 15 x - 50

= \int x^{4} + 5 x^{3} + 5 x^{2} + 15 x - 50 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^{4} d x + \int 5 x^{3} d x + \int 5 x^{2} d x + \int 15 x d x - \int 50 d x

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

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

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

\int 15 x d x = \frac{15 x ^{2}}{2}

\int 50 d x = 50 x

= \frac{x ^{5}}{5} + \frac{5 x ^{4}}{4} + \frac{5 x ^{3}}{3} + \frac{15 x ^{2}}{2} - 50 x

定数を解答に追加する

= \frac{x ^{5}}{5} + \frac{5 x ^{4}}{4} + \frac{5 x ^{3}}{3} + \frac{15 x ^{2}}{2} - 50 x + C