integral of (7-2x)^{11}

解

\int (7 - 2 x)^{11} d x

7^{11} x - 3107227739 x^{2} + \frac{8877793540 x ^{3}}{3} - 1902384330 x^{4} + 869661408 x^{5} - 289887136 x^{6} + 70992768 x^{7} - 12677280 x^{8} + \frac{4829440 x ^{9}}{3} - 137984 x^{10} + 7168 x^{11} - \frac{512 x ^{12}}{3} + C

解答ステップ

\int (7 - 2 x)^{11} d x

拡張

(7 - 2 x)^{11} : 7^{11} - 6214455478 x + 8877793540 x^{2} - 7609537320 x^{3} + 4348307040 x^{4} - 1739322816 x^{5} + 496949376 x^{6} - 101418240 x^{7} + 14488320 x^{8} - 1379840 x^{9} + 78848 x^{10} - 2048 x^{11}

= \int 7^{11} - 6214455478 x + 8877793540 x^{2} - 7609537320 x^{3} + 4348307040 x^{4} - 1739322816 x^{5} + 496949376 x^{6} - 101418240 x^{7} + 14488320 x^{8} - 1379840 x^{9} + 78848 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 7^{11} d x - \int 6214455478 x d x + \int 8877793540 x^{2} d x - \int 7609537320 x^{3} d x + \int 4348307040 x^{4} d x - \int 1739322816 x^{5} d x + \int 496949376 x^{6} d x - \int 101418240 x^{7} d x + \int 14488320 x^{8} d x - \int 1379840 x^{9} d x + \int 78848 x^{10} d x - \int 2048 x^{11} d x

\int 7^{11} d x = 7^{11} x

\int 6214455478 x d x = 3107227739 x^{2}

\int 8877793540 x^{2} d x = \frac{8877793540 x ^{3}}{3}

\int 7609537320 x^{3} d x = 1902384330 x^{4}

\int 4348307040 x^{4} d x = 869661408 x^{5}

\int 1739322816 x^{5} d x = 289887136 x^{6}

\int 496949376 x^{6} d x = 70992768 x^{7}

\int 101418240 x^{7} d x = 12677280 x^{8}

\int 14488320 x^{8} d x = \frac{4829440 x ^{9}}{3}

\int 1379840 x^{9} d x = 137984 x^{10}

\int 78848 x^{10} d x = 7168 x^{11}

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

= 7^{11} x - 3107227739 x^{2} + \frac{8877793540 x ^{3}}{3} - 1902384330 x^{4} + 869661408 x^{5} - 289887136 x^{6} + 70992768 x^{7} - 12677280 x^{8} + \frac{4829440 x ^{9}}{3} - 137984 x^{10} + 7168 x^{11} - \frac{512 x ^{12}}{3}

定数を解答に追加する

= 7^{11} x - 3107227739 x^{2} + \frac{8877793540 x ^{3}}{3} - 1902384330 x^{4} + 869661408 x^{5} - 289887136 x^{6} + 70992768 x^{7} - 12677280 x^{8} + \frac{4829440 x ^{9}}{3} - 137984 x^{10} + 7168 x^{11} - \frac{512 x ^{12}}{3} + C