integral of (x^2+4x)^5(x+2)

解

\int (x^{2} + 4 x)^{5} (x + 2) d x

\frac{x ^{12}}{12} + 2 x^{11} + 20 x^{10} + \frac{320 x ^{9}}{3} + 320 x^{8} + 512 x^{7} + \frac{1024 x ^{6}}{3} + C

解答ステップ

\int (x^{2} + 4 x)^{5} (x + 2) d x

拡張

(x^{2} + 4 x)^{5} (x + 2) : x^{11} + 22 x^{10} + 200 x^{9} + 960 x^{8} + 2560 x^{7} + 3584 x^{6} + 2048 x^{5}

= \int x^{11} + 22 x^{10} + 200 x^{9} + 960 x^{8} + 2560 x^{7} + 3584 x^{6} + 2048 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^{11} d x + \int 22 x^{10} d x + \int 200 x^{9} d x + \int 960 x^{8} d x + \int 2560 x^{7} d x + \int 3584 x^{6} d x + \int 2048 x^{5} d x

\int x^{11} d x = \frac{x ^{12}}{12}

\int 22 x^{10} d x = 2 x^{11}

\int 200 x^{9} d x = 20 x^{10}

\int 960 x^{8} d x = \frac{320 x ^{9}}{3}

\int 2560 x^{7} d x = 320 x^{8}

\int 3584 x^{6} d x = 512 x^{7}

\int 2048 x^{5} d x = \frac{1024 x ^{6}}{3}

= \frac{x ^{12}}{12} + 2 x^{11} + 20 x^{10} + \frac{320 x ^{9}}{3} + 320 x^{8} + 512 x^{7} + \frac{1024 x ^{6}}{3}

定数を解答に追加する

= \frac{x ^{12}}{12} + 2 x^{11} + 20 x^{10} + \frac{320 x ^{9}}{3} + 320 x^{8} + 512 x^{7} + \frac{1024 x ^{6}}{3} + C