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

解

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

\frac{x ^{8}}{4} - x^{7} + \frac{9 x ^{6}}{2} - 10 x^{5} + \frac{91 x ^{4}}{4} - 30 x^{3} + \frac{81 x ^{2}}{2} - 27 x + C

解答ステップ

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

拡張

(2 x - 1) (x^{2} - x + 3)^{3} : 2 x^{7} - 7 x^{6} + 27 x^{5} - 50 x^{4} + 91 x^{3} - 90 x^{2} + 81 x - 27

= \int 2 x^{7} - 7 x^{6} + 27 x^{5} - 50 x^{4} + 91 x^{3} - 90 x^{2} + 81 x - 27 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 2 x^{7} d x - \int 7 x^{6} d x + \int 27 x^{5} d x - \int 50 x^{4} d x + \int 91 x^{3} d x - \int 90 x^{2} d x + \int 81 x d x - \int 27 d x

\int 2 x^{7} d x = \frac{x ^{8}}{4}

\int 7 x^{6} d x = x^{7}

\int 27 x^{5} d x = \frac{9 x ^{6}}{2}

\int 50 x^{4} d x = 10 x^{5}

\int 91 x^{3} d x = \frac{91 x ^{4}}{4}

\int 90 x^{2} d x = 30 x^{3}

\int 81 x d x = \frac{81 x ^{2}}{2}

\int 27 d x = 27 x

= \frac{x ^{8}}{4} - x^{7} + \frac{9 x ^{6}}{2} - 10 x^{5} + \frac{91 x ^{4}}{4} - 30 x^{3} + \frac{81 x ^{2}}{2} - 27 x

定数を解答に追加する

= \frac{x ^{8}}{4} - x^{7} + \frac{9 x ^{6}}{2} - 10 x^{5} + \frac{91 x ^{4}}{4} - 30 x^{3} + \frac{81 x ^{2}}{2} - 27 x + C