integral of (x^2+x)^{12}(2x+1)

解

\int (x^{2} + x)^{12} (2 x + 1) d x

\frac{x ^{26}}{13} + x^{25} + 6 x^{24} + 22 x^{23} + 55 x^{22} + 99 x^{21} + 132 x^{20} + 132 x^{19} + 99 x^{18} + 55 x^{17} + 22 x^{16} + 6 x^{15} + x^{14} + \frac{x ^{13}}{13} + C

解答ステップ

\int (x^{2} + x)^{12} (2 x + 1) d x

拡張

(x^{2} + x)^{12} (2 x + 1) : 2 x^{25} + 25 x^{24} + 144 x^{23} + 506 x^{22} + 1210 x^{21} + 2079 x^{20} + 2640 x^{19} + 2508 x^{18} + 1782 x^{17} + 935 x^{16} + 352 x^{15} + 90 x^{14} + 14 x^{13} + x^{12}

= \int 2 x^{25} + 25 x^{24} + 144 x^{23} + 506 x^{22} + 1210 x^{21} + 2079 x^{20} + 2640 x^{19} + 2508 x^{18} + 1782 x^{17} + 935 x^{16} + 352 x^{15} + 90 x^{14} + 14 x^{13} + x^{12} 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 2 x^{25} d x + \int 25 x^{24} d x + \int 144 x^{23} d x + \int 506 x^{22} d x + \int 1210 x^{21} d x + \int 2079 x^{20} d x + \int 2640 x^{19} d x + \int 2508 x^{18} d x + \int 1782 x^{17} d x + \int 935 x^{16} d x + \int 352 x^{15} d x + \int 90 x^{14} d x + \int 14 x^{13} d x + \int x^{12} d x

\int 2 x^{25} d x = \frac{x ^{26}}{13}

\int 25 x^{24} d x = x^{25}

\int 144 x^{23} d x = 6 x^{24}

\int 506 x^{22} d x = 22 x^{23}

\int 1210 x^{21} d x = 55 x^{22}

\int 2079 x^{20} d x = 99 x^{21}

\int 2640 x^{19} d x = 132 x^{20}

\int 2508 x^{18} d x = 132 x^{19}

\int 1782 x^{17} d x = 99 x^{18}

\int 935 x^{16} d x = 55 x^{17}

\int 352 x^{15} d x = 22 x^{16}

\int 90 x^{14} d x = 6 x^{15}

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

\int x^{12} d x = \frac{x ^{13}}{13}

= \frac{x ^{26}}{13} + x^{25} + 6 x^{24} + 22 x^{23} + 55 x^{22} + 99 x^{21} + 132 x^{20} + 132 x^{19} + 99 x^{18} + 55 x^{17} + 22 x^{16} + 6 x^{15} + x^{14} + \frac{x ^{13}}{13}

定数を解答に追加する

= \frac{x ^{26}}{13} + x^{25} + 6 x^{24} + 22 x^{23} + 55 x^{22} + 99 x^{21} + 132 x^{20} + 132 x^{19} + 99 x^{18} + 55 x^{17} + 22 x^{16} + 6 x^{15} + x^{14} + \frac{x ^{13}}{13} + C