integral of x^2(2x+1)^9

解

\int x^{2} (2 x + 1)^{9} d x

\frac{128 x ^{12}}{3} + \frac{2304 x ^{11}}{11} + \frac{2304 x ^{10}}{5} + \frac{1792 x ^{9}}{3} + 504 x^{8} + 288 x^{7} + 112 x^{6} + \frac{144 x ^{5}}{5} + \frac{9 x ^{4}}{2} + \frac{x ^{3}}{3} + C

解答ステップ

\int x^{2} (2 x + 1)^{9} d x

拡張

x^{2} (2 x + 1)^{9} : 512 x^{11} + 2304 x^{10} + 4608 x^{9} + 5376 x^{8} + 4032 x^{7} + 2016 x^{6} + 672 x^{5} + 144 x^{4} + 18 x^{3} + x^{2}

= \int 512 x^{11} + 2304 x^{10} + 4608 x^{9} + 5376 x^{8} + 4032 x^{7} + 2016 x^{6} + 672 x^{5} + 144 x^{4} + 18 x^{3} + 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 512 x^{11} d x + \int 2304 x^{10} d x + \int 4608 x^{9} d x + \int 5376 x^{8} d x + \int 4032 x^{7} d x + \int 2016 x^{6} d x + \int 672 x^{5} d x + \int 144 x^{4} d x + \int 18 x^{3} d x + \int x^{2} d x

\int 512 x^{11} d x = \frac{128 x ^{12}}{3}

\int 2304 x^{10} d x = \frac{2304 x ^{11}}{11}

\int 4608 x^{9} d x = \frac{2304 x ^{10}}{5}

\int 5376 x^{8} d x = \frac{1792 x ^{9}}{3}

\int 4032 x^{7} d x = 504 x^{8}

\int 2016 x^{6} d x = 288 x^{7}

\int 672 x^{5} d x = 112 x^{6}

\int 144 x^{4} d x = \frac{144 x ^{5}}{5}

\int 18 x^{3} d x = \frac{9 x ^{4}}{2}

\int x^{2} d x = \frac{x ^{3}}{3}

= \frac{128 x ^{12}}{3} + \frac{2304 x ^{11}}{11} + \frac{2304 x ^{10}}{5} + \frac{1792 x ^{9}}{3} + 504 x^{8} + 288 x^{7} + 112 x^{6} + \frac{144 x ^{5}}{5} + \frac{9 x ^{4}}{2} + \frac{x ^{3}}{3}

定数を解答に追加する

= \frac{128 x ^{12}}{3} + \frac{2304 x ^{11}}{11} + \frac{2304 x ^{10}}{5} + \frac{1792 x ^{9}}{3} + 504 x^{8} + 288 x^{7} + 112 x^{6} + \frac{144 x ^{5}}{5} + \frac{9 x ^{4}}{2} + \frac{x ^{3}}{3} + C