integral of (x-2)(x-3)^6

解

\int (x - 2) (x - 3)^{6} d x

\frac{x ^{8}}{8} - \frac{20 x ^{7}}{7} + \frac{57 x ^{6}}{2} - 162 x^{5} + \frac{2295 x ^{4}}{4} - 1296 x^{3} + \frac{3645 x ^{2}}{2} - 1458 x + C

解答ステップ

\int (x - 2) (x - 3)^{6} d x

拡張

(x - 2) (x - 3)^{6} : x^{7} - 20 x^{6} + 171 x^{5} - 810 x^{4} + 2295 x^{3} - 3888 x^{2} + 3645 x - 1458

= \int x^{7} - 20 x^{6} + 171 x^{5} - 810 x^{4} + 2295 x^{3} - 3888 x^{2} + 3645 x - 1458 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^{7} d x - \int 20 x^{6} d x + \int 171 x^{5} d x - \int 810 x^{4} d x + \int 2295 x^{3} d x - \int 3888 x^{2} d x + \int 3645 x d x - \int 1458 d x

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

\int 20 x^{6} d x = \frac{20 x ^{7}}{7}

\int 171 x^{5} d x = \frac{57 x ^{6}}{2}

\int 810 x^{4} d x = 162 x^{5}

\int 2295 x^{3} d x = \frac{2295 x ^{4}}{4}

\int 3888 x^{2} d x = 1296 x^{3}

\int 3645 x d x = \frac{3645 x ^{2}}{2}

\int 1458 d x = 1458 x

= \frac{x ^{8}}{8} - \frac{20 x ^{7}}{7} + \frac{57 x ^{6}}{2} - 162 x^{5} + \frac{2295 x ^{4}}{4} - 1296 x^{3} + \frac{3645 x ^{2}}{2} - 1458 x

定数を解答に追加する

= \frac{x ^{8}}{8} - \frac{20 x ^{7}}{7} + \frac{57 x ^{6}}{2} - 162 x^{5} + \frac{2295 x ^{4}}{4} - 1296 x^{3} + \frac{3645 x ^{2}}{2} - 1458 x + C