integral of (2x-5)(x-3)^9

解

\int (2 x - 5) (x - 3)^{9} d x

\frac{2 x ^{11}}{11} - \frac{59 x ^{10}}{10} + 87 x^{9} - \frac{1539 x ^{8}}{2} + 4536 x^{7} - 18711 x^{6} + \frac{275562 x ^{5}}{5} - 115911 x^{4} + 170586 x^{3} - \frac{334611 x ^{2}}{2} + 98415 x + C

解答ステップ

\int (2 x - 5) (x - 3)^{9} d x

拡張

(2 x - 5) (x - 3)^{9} : 2 x^{10} - 59 x^{9} + 783 x^{8} - 6156 x^{7} + 31752 x^{6} - 112266 x^{5} + 275562 x^{4} - 463644 x^{3} + 511758 x^{2} - 334611 x + 98415

= \int 2 x^{10} - 59 x^{9} + 783 x^{8} - 6156 x^{7} + 31752 x^{6} - 112266 x^{5} + 275562 x^{4} - 463644 x^{3} + 511758 x^{2} - 334611 x + 98415 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^{10} d x - \int 59 x^{9} d x + \int 783 x^{8} d x - \int 6156 x^{7} d x + \int 31752 x^{6} d x - \int 112266 x^{5} d x + \int 275562 x^{4} d x - \int 463644 x^{3} d x + \int 511758 x^{2} d x - \int 334611 x d x + \int 98415 d x

\int 2 x^{10} d x = \frac{2 x ^{11}}{11}

\int 59 x^{9} d x = \frac{59 x ^{10}}{10}

\int 783 x^{8} d x = 87 x^{9}

\int 6156 x^{7} d x = \frac{1539 x ^{8}}{2}

\int 31752 x^{6} d x = 4536 x^{7}

\int 112266 x^{5} d x = 18711 x^{6}

\int 275562 x^{4} d x = \frac{275562 x ^{5}}{5}

\int 463644 x^{3} d x = 115911 x^{4}

\int 511758 x^{2} d x = 170586 x^{3}

\int 334611 x d x = \frac{334611 x ^{2}}{2}

\int 98415 d x = 98415 x

= \frac{2 x ^{11}}{11} - \frac{59 x ^{10}}{10} + 87 x^{9} - \frac{1539 x ^{8}}{2} + 4536 x^{7} - 18711 x^{6} + \frac{275562 x ^{5}}{5} - 115911 x^{4} + 170586 x^{3} - \frac{334611 x ^{2}}{2} + 98415 x

定数を解答に追加する

= \frac{2 x ^{11}}{11} - \frac{59 x ^{10}}{10} + 87 x^{9} - \frac{1539 x ^{8}}{2} + 4536 x^{7} - 18711 x^{6} + \frac{275562 x ^{5}}{5} - 115911 x^{4} + 170586 x^{3} - \frac{334611 x ^{2}}{2} + 98415 x + C