integral of (5-2x)^{11}

解

\int (5 - 2 x)^{11} d x

48828125 x - 107421875 x^{2} + \frac{429687500 x ^{3}}{3} - 128906250 x^{4} + 82500000 x^{5} - 38500000 x^{6} + 13200000 x^{7} - 3300000 x^{8} + \frac{1760000 x ^{9}}{3} - 70400 x^{10} + 5120 x^{11} - \frac{512 x ^{12}}{3} + C

解答ステップ

\int (5 - 2 x)^{11} d x

拡張

(5 - 2 x)^{11} : 48828125 - 214843750 x + 429687500 x^{2} - 515625000 x^{3} + 412500000 x^{4} - 231000000 x^{5} + 92400000 x^{6} - 26400000 x^{7} + 5280000 x^{8} - 704000 x^{9} + 56320 x^{10} - 2048 x^{11}

= \int 48828125 - 214843750 x + 429687500 x^{2} - 515625000 x^{3} + 412500000 x^{4} - 231000000 x^{5} + 92400000 x^{6} - 26400000 x^{7} + 5280000 x^{8} - 704000 x^{9} + 56320 x^{10} - 2048 x^{11} 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 48828125 d x - \int 214843750 x d x + \int 429687500 x^{2} d x - \int 515625000 x^{3} d x + \int 412500000 x^{4} d x - \int 231000000 x^{5} d x + \int 92400000 x^{6} d x - \int 26400000 x^{7} d x + \int 5280000 x^{8} d x - \int 704000 x^{9} d x + \int 56320 x^{10} d x - \int 2048 x^{11} d x

\int 48828125 d x = 48828125 x

\int 214843750 x d x = 107421875 x^{2}

\int 429687500 x^{2} d x = \frac{429687500 x ^{3}}{3}

\int 515625000 x^{3} d x = 128906250 x^{4}

\int 412500000 x^{4} d x = 82500000 x^{5}

\int 231000000 x^{5} d x = 38500000 x^{6}

\int 92400000 x^{6} d x = 13200000 x^{7}

\int 26400000 x^{7} d x = 3300000 x^{8}

\int 5280000 x^{8} d x = \frac{1760000 x ^{9}}{3}

\int 704000 x^{9} d x = 70400 x^{10}

\int 56320 x^{10} d x = 5120 x^{11}

\int 2048 x^{11} d x = \frac{512 x ^{12}}{3}

= 48828125 x - 107421875 x^{2} + \frac{429687500 x ^{3}}{3} - 128906250 x^{4} + 82500000 x^{5} - 38500000 x^{6} + 13200000 x^{7} - 3300000 x^{8} + \frac{1760000 x ^{9}}{3} - 70400 x^{10} + 5120 x^{11} - \frac{512 x ^{12}}{3}

定数を解答に追加する

= 48828125 x - 107421875 x^{2} + \frac{429687500 x ^{3}}{3} - 128906250 x^{4} + 82500000 x^{5} - 38500000 x^{6} + 13200000 x^{7} - 3300000 x^{8} + \frac{1760000 x ^{9}}{3} - 70400 x^{10} + 5120 x^{11} - \frac{512 x ^{12}}{3} + C