integral of (x^2+x)^{13}(2x+1)

解

\int (x^{2} + x)^{13} (2 x + 1) d x

\frac{x ^{28}}{14} + x^{27} + \frac{13 x ^{26}}{2} + 26 x^{25} + \frac{143 x ^{24}}{2} + 143 x^{23} + \frac{429 x ^{22}}{2} + \frac{1716 x ^{21}}{7} + \frac{429 x ^{20}}{2} + 143 x^{19} + \frac{143 x ^{18}}{2} + 26 x^{17} + \frac{13 x ^{16}}{2} + x^{15} + \frac{x ^{14}}{14} + C

解答ステップ

\int (x^{2} + x)^{13} (2 x + 1) d x

拡張

(x^{2} + x)^{13} (2 x + 1) : 2 x^{27} + 27 x^{26} + 169 x^{25} + 650 x^{24} + 1716 x^{23} + 3289 x^{22} + 4719 x^{21} + 5148 x^{20} + 4290 x^{19} + 2717 x^{18} + 1287 x^{17} + 442 x^{16} + 104 x^{15} + 15 x^{14} + x^{13}

= \int 2 x^{27} + 27 x^{26} + 169 x^{25} + 650 x^{24} + 1716 x^{23} + 3289 x^{22} + 4719 x^{21} + 5148 x^{20} + 4290 x^{19} + 2717 x^{18} + 1287 x^{17} + 442 x^{16} + 104 x^{15} + 15 x^{14} + x^{13} 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^{27} d x + \int 27 x^{26} d x + \int 169 x^{25} d x + \int 650 x^{24} d x + \int 1716 x^{23} d x + \int 3289 x^{22} d x + \int 4719 x^{21} d x + \int 5148 x^{20} d x + \int 4290 x^{19} d x + \int 2717 x^{18} d x + \int 1287 x^{17} d x + \int 442 x^{16} d x + \int 104 x^{15} d x + \int 15 x^{14} d x + \int x^{13} d x

\int 2 x^{27} d x = \frac{x ^{28}}{14}

\int 27 x^{26} d x = x^{27}

\int 169 x^{25} d x = \frac{13 x ^{26}}{2}

\int 650 x^{24} d x = 26 x^{25}

\int 1716 x^{23} d x = \frac{143 x ^{24}}{2}

\int 3289 x^{22} d x = 143 x^{23}

\int 4719 x^{21} d x = \frac{429 x ^{22}}{2}

\int 5148 x^{20} d x = \frac{1716 x ^{21}}{7}

\int 4290 x^{19} d x = \frac{429 x ^{20}}{2}

\int 2717 x^{18} d x = 143 x^{19}

\int 1287 x^{17} d x = \frac{143 x ^{18}}{2}

\int 442 x^{16} d x = 26 x^{17}

\int 104 x^{15} d x = \frac{13 x ^{16}}{2}

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

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

= \frac{x ^{28}}{14} + x^{27} + \frac{13 x ^{26}}{2} + 26 x^{25} + \frac{143 x ^{24}}{2} + 143 x^{23} + \frac{429 x ^{22}}{2} + \frac{1716 x ^{21}}{7} + \frac{429 x ^{20}}{2} + 143 x^{19} + \frac{143 x ^{18}}{2} + 26 x^{17} + \frac{13 x ^{16}}{2} + x^{15} + \frac{x ^{14}}{14}

定数を解答に追加する

= \frac{x ^{28}}{14} + x^{27} + \frac{13 x ^{26}}{2} + 26 x^{25} + \frac{143 x ^{24}}{2} + 143 x^{23} + \frac{429 x ^{22}}{2} + \frac{1716 x ^{21}}{7} + \frac{429 x ^{20}}{2} + 143 x^{19} + \frac{143 x ^{18}}{2} + 26 x^{17} + \frac{13 x ^{16}}{2} + x^{15} + \frac{x ^{14}}{14} + C