integral of x^3(x^4-6)^7

解

\int x^{3} (x^{4} - 6)^{7} d x

\frac{x ^{32}}{32} - \frac{3 x ^{28}}{2} + \frac{63 x ^{24}}{2} - 378 x^{20} + 2835 x^{16} - 13608 x^{12} + 40824 x^{8} - 69984 x^{4} + C

解答ステップ

\int x^{3} (x^{4} - 6)^{7} d x

拡張

x^{3} (x^{4} - 6)^{7} : x^{31} - 42 x^{27} + 756 x^{23} - 7560 x^{19} + 45360 x^{15} - 163296 x^{11} + 326592 x^{7} - 279936 x^{3}

= \int x^{31} - 42 x^{27} + 756 x^{23} - 7560 x^{19} + 45360 x^{15} - 163296 x^{11} + 326592 x^{7} - 279936 x^{3} 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^{31} d x - \int 42 x^{27} d x + \int 756 x^{23} d x - \int 7560 x^{19} d x + \int 45360 x^{15} d x - \int 163296 x^{11} d x + \int 326592 x^{7} d x - \int 279936 x^{3} d x

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

\int 42 x^{27} d x = \frac{3 x ^{28}}{2}

\int 756 x^{23} d x = \frac{63 x ^{24}}{2}

\int 7560 x^{19} d x = 378 x^{20}

\int 45360 x^{15} d x = 2835 x^{16}

\int 163296 x^{11} d x = 13608 x^{12}

\int 326592 x^{7} d x = 40824 x^{8}

\int 279936 x^{3} d x = 69984 x^{4}

= \frac{x ^{32}}{32} - \frac{3 x ^{28}}{2} + \frac{63 x ^{24}}{2} - 378 x^{20} + 2835 x^{16} - 13608 x^{12} + 40824 x^{8} - 69984 x^{4}

定数を解答に追加する

= \frac{x ^{32}}{32} - \frac{3 x ^{28}}{2} + \frac{63 x ^{24}}{2} - 378 x^{20} + 2835 x^{16} - 13608 x^{12} + 40824 x^{8} - 69984 x^{4} + C