integral of x^2(7x^3-3)^5

解

\int x^{2} (7 x^{3} - 3)^{5} d x

\frac{16807 x ^{18}}{18} - 2401 x^{15} + \frac{5145 x ^{12}}{2} - 1470 x^{9} + \frac{945 x ^{6}}{2} - 81 x^{3} + C

解答ステップ

\int x^{2} (7 x^{3} - 3)^{5} d x

拡張

x^{2} (7 x^{3} - 3)^{5} : 16807 x^{17} - 36015 x^{14} + 30870 x^{11} - 13230 x^{8} + 2835 x^{5} - 243 x^{2}

= \int 16807 x^{17} - 36015 x^{14} + 30870 x^{11} - 13230 x^{8} + 2835 x^{5} - 243 x^{2} 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 16807 x^{17} d x - \int 36015 x^{14} d x + \int 30870 x^{11} d x - \int 13230 x^{8} d x + \int 2835 x^{5} d x - \int 243 x^{2} d x

\int 16807 x^{17} d x = \frac{16807 x ^{18}}{18}

\int 36015 x^{14} d x = 2401 x^{15}

\int 30870 x^{11} d x = \frac{5145 x ^{12}}{2}

\int 13230 x^{8} d x = 1470 x^{9}

\int 2835 x^{5} d x = \frac{945 x ^{6}}{2}

\int 243 x^{2} d x = 81 x^{3}

= \frac{16807 x ^{18}}{18} - 2401 x^{15} + \frac{5145 x ^{12}}{2} - 1470 x^{9} + \frac{945 x ^{6}}{2} - 81 x^{3}

定数を解答に追加する

= \frac{16807 x ^{18}}{18} - 2401 x^{15} + \frac{5145 x ^{12}}{2} - 1470 x^{9} + \frac{945 x ^{6}}{2} - 81 x^{3} + C