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

解

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

2 (\frac{x ^{20}}{20} + x^{18} + 9 x^{16} + 48 x^{14} + 168 x^{12} + \frac{2016 x ^{10}}{5} + 672 x^{8} + 768 x^{6} + 576 x^{4} + 256 x^{2}) + C

解答ステップ

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

拡張

x (x^{2} + 2)^{9} : x^{19} + 18 x^{17} + 144 x^{15} + 672 x^{13} + 2016 x^{11} + 4032 x^{9} + 5376 x^{7} + 4608 x^{5} + 2304 x^{3} + 512 x

= 2 \cdot \int x^{19} + 18 x^{17} + 144 x^{15} + 672 x^{13} + 2016 x^{11} + 4032 x^{9} + 5376 x^{7} + 4608 x^{5} + 2304 x^{3} + 512 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

= 2 (\int x^{19} d x + \int 18 x^{17} d x + \int 144 x^{15} d x + \int 672 x^{13} d x + \int 2016 x^{11} d x + \int 4032 x^{9} d x + \int 5376 x^{7} d x + \int 4608 x^{5} d x + \int 2304 x^{3} d x + \int 512 x d x)

\int x^{19} d x = \frac{x ^{20}}{20}

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

\int 144 x^{15} d x = 9 x^{16}

\int 672 x^{13} d x = 48 x^{14}

\int 2016 x^{11} d x = 168 x^{12}

\int 4032 x^{9} d x = \frac{2016 x ^{10}}{5}

\int 5376 x^{7} d x = 672 x^{8}

\int 4608 x^{5} d x = 768 x^{6}

\int 2304 x^{3} d x = 576 x^{4}

\int 512 x d x = 256 x^{2}

= 2 (\frac{x ^{20}}{20} + x^{18} + 9 x^{16} + 48 x^{14} + 168 x^{12} + \frac{2016 x ^{10}}{5} + 672 x^{8} + 768 x^{6} + 576 x^{4} + 256 x^{2})

定数を解答に追加する

= 2 (\frac{x ^{20}}{20} + x^{18} + 9 x^{16} + 48 x^{14} + 168 x^{12} + \frac{2016 x ^{10}}{5} + 672 x^{8} + 768 x^{6} + 576 x^{4} + 256 x^{2}) + C