integral of x(x+8)^6

解

\int x (x + 8)^{6} d x

\frac{x ^{8}}{8} + \frac{48 x ^{7}}{7} + 160 x^{6} + 2048 x^{5} + 15360 x^{4} + 65536 x^{3} + 131072 x^{2} + C

解答ステップ

\int x (x + 8)^{6} d x

拡張

x (x + 8)^{6} : x^{7} + 48 x^{6} + 960 x^{5} + 10240 x^{4} + 61440 x^{3} + 196608 x^{2} + 262144 x

= \int x^{7} + 48 x^{6} + 960 x^{5} + 10240 x^{4} + 61440 x^{3} + 196608 x^{2} + 262144 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 48 x^{6} d x + \int 960 x^{5} d x + \int 10240 x^{4} d x + \int 61440 x^{3} d x + \int 196608 x^{2} d x + \int 262144 x d x

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

\int 48 x^{6} d x = \frac{48 x ^{7}}{7}

\int 960 x^{5} d x = 160 x^{6}

\int 10240 x^{4} d x = 2048 x^{5}

\int 61440 x^{3} d x = 15360 x^{4}

\int 196608 x^{2} d x = 65536 x^{3}

\int 262144 x d x = 131072 x^{2}

= \frac{x ^{8}}{8} + \frac{48 x ^{7}}{7} + 160 x^{6} + 2048 x^{5} + 15360 x^{4} + 65536 x^{3} + 131072 x^{2}

定数を解答に追加する

= \frac{x ^{8}}{8} + \frac{48 x ^{7}}{7} + 160 x^{6} + 2048 x^{5} + 15360 x^{4} + 65536 x^{3} + 131072 x^{2} + C