integral of (x^2+2)(x^3+6x)^5

解

\int (x^{2} + 2) (x^{3} + 6 x)^{5} d x

\frac{x ^{18}}{18} + 2 x^{16} + 30 x^{14} + 240 x^{12} + 1080 x^{10} + 2592 x^{8} + 2592 x^{6} + C

解答ステップ

\int (x^{2} + 2) (x^{3} + 6 x)^{5} d x

拡張

(x^{2} + 2) (x^{3} + 6 x)^{5} : x^{17} + 32 x^{15} + 420 x^{13} + 2880 x^{11} + 10800 x^{9} + 20736 x^{7} + 15552 x^{5}

= \int x^{17} + 32 x^{15} + 420 x^{13} + 2880 x^{11} + 10800 x^{9} + 20736 x^{7} + 15552 x^{5} 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^{17} d x + \int 32 x^{15} d x + \int 420 x^{13} d x + \int 2880 x^{11} d x + \int 10800 x^{9} d x + \int 20736 x^{7} d x + \int 15552 x^{5} d x

\int x^{17} d x = \frac{x ^{18}}{18}

\int 32 x^{15} d x = 2 x^{16}

\int 420 x^{13} d x = 30 x^{14}

\int 2880 x^{11} d x = 240 x^{12}

\int 10800 x^{9} d x = 1080 x^{10}

\int 20736 x^{7} d x = 2592 x^{8}

\int 15552 x^{5} d x = 2592 x^{6}

= \frac{x ^{18}}{18} + 2 x^{16} + 30 x^{14} + 240 x^{12} + 1080 x^{10} + 2592 x^{8} + 2592 x^{6}

定数を解答に追加する

= \frac{x ^{18}}{18} + 2 x^{16} + 30 x^{14} + 240 x^{12} + 1080 x^{10} + 2592 x^{8} + 2592 x^{6} + C