integral of (x^9-3x)^7

解

\int (x^{9} - 3 x)^{7} d x

\frac{x ^{64}}{64} - \frac{3 x ^{56}}{8} + \frac{63 x ^{48}}{16} - \frac{189 x ^{40}}{8} + \frac{2835 x ^{32}}{32} - \frac{1701 x ^{24}}{8} + \frac{5103 x ^{16}}{16} - \frac{2187 x ^{8}}{8} + C

解答ステップ

\int (x^{9} - 3 x)^{7} d x

拡張

(x^{9} - 3 x)^{7} : x^{63} - 21 x^{55} + 189 x^{47} - 945 x^{39} + 2835 x^{31} - 5103 x^{23} + 5103 x^{15} - 2187 x^{7}

= \int x^{63} - 21 x^{55} + 189 x^{47} - 945 x^{39} + 2835 x^{31} - 5103 x^{23} + 5103 x^{15} - 2187 x^{7} 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^{63} d x - \int 21 x^{55} d x + \int 189 x^{47} d x - \int 945 x^{39} d x + \int 2835 x^{31} d x - \int 5103 x^{23} d x + \int 5103 x^{15} d x - \int 2187 x^{7} d x

\int x^{63} d x = \frac{x ^{64}}{64}

\int 21 x^{55} d x = \frac{3 x ^{56}}{8}

\int 189 x^{47} d x = \frac{63 x ^{48}}{16}

\int 945 x^{39} d x = \frac{189 x ^{40}}{8}

\int 2835 x^{31} d x = \frac{2835 x ^{32}}{32}

\int 5103 x^{23} d x = \frac{1701 x ^{24}}{8}

\int 5103 x^{15} d x = \frac{5103 x ^{16}}{16}

\int 2187 x^{7} d x = \frac{2187 x ^{8}}{8}

= \frac{x ^{64}}{64} - \frac{3 x ^{56}}{8} + \frac{63 x ^{48}}{16} - \frac{189 x ^{40}}{8} + \frac{2835 x ^{32}}{32} - \frac{1701 x ^{24}}{8} + \frac{5103 x ^{16}}{16} - \frac{2187 x ^{8}}{8}

定数を解答に追加する

= \frac{x ^{64}}{64} - \frac{3 x ^{56}}{8} + \frac{63 x ^{48}}{16} - \frac{189 x ^{40}}{8} + \frac{2835 x ^{32}}{32} - \frac{1701 x ^{24}}{8} + \frac{5103 x ^{16}}{16} - \frac{2187 x ^{8}}{8} + C