integral of x(x+4)^6

解

\int x (x + 4)^{6} d x

\frac{x ^{8}}{8} + \frac{24 x ^{7}}{7} + 40 x^{6} + 256 x^{5} + 960 x^{4} + 2048 x^{3} + 2048 x^{2} + C

解答ステップ

\int x (x + 4)^{6} d x

拡張

x (x + 4)^{6} : x^{7} + 24 x^{6} + 240 x^{5} + 1280 x^{4} + 3840 x^{3} + 6144 x^{2} + 4096 x

= \int x^{7} + 24 x^{6} + 240 x^{5} + 1280 x^{4} + 3840 x^{3} + 6144 x^{2} + 4096 x 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^{7} d x + \int 24 x^{6} d x + \int 240 x^{5} d x + \int 1280 x^{4} d x + \int 3840 x^{3} d x + \int 6144 x^{2} d x + \int 4096 x d x

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

\int 24 x^{6} d x = \frac{24 x ^{7}}{7}

\int 240 x^{5} d x = 40 x^{6}

\int 1280 x^{4} d x = 256 x^{5}

\int 3840 x^{3} d x = 960 x^{4}

\int 6144 x^{2} d x = 2048 x^{3}

\int 4096 x d x = 2048 x^{2}

= \frac{x ^{8}}{8} + \frac{24 x ^{7}}{7} + 40 x^{6} + 256 x^{5} + 960 x^{4} + 2048 x^{3} + 2048 x^{2}

定数を解答に追加する

= \frac{x ^{8}}{8} + \frac{24 x ^{7}}{7} + 40 x^{6} + 256 x^{5} + 960 x^{4} + 2048 x^{3} + 2048 x^{2} + C