integral of (x^3+x)^9

解

\int (x^{3} + x)^{9} d x

\frac{x ^{28}}{28} + \frac{9 x ^{26}}{26} + \frac{3 x ^{24}}{2} + \frac{42 x ^{22}}{11} + \frac{63 x ^{20}}{10} + 7 x^{18} + \frac{21 x ^{16}}{4} + \frac{18 x ^{14}}{7} + \frac{3 x ^{12}}{4} + \frac{x ^{10}}{10} + C

解答ステップ

\int (x^{3} + x)^{9} d x

拡張

(x^{3} + x)^{9} : x^{27} + 9 x^{25} + 36 x^{23} + 84 x^{21} + 126 x^{19} + 126 x^{17} + 84 x^{15} + 36 x^{13} + 9 x^{11} + x^{9}

= \int x^{27} + 9 x^{25} + 36 x^{23} + 84 x^{21} + 126 x^{19} + 126 x^{17} + 84 x^{15} + 36 x^{13} + 9 x^{11} + x^{9} 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^{27} d x + \int 9 x^{25} d x + \int 36 x^{23} d x + \int 84 x^{21} d x + \int 126 x^{19} d x + \int 126 x^{17} d x + \int 84 x^{15} d x + \int 36 x^{13} d x + \int 9 x^{11} d x + \int x^{9} d x

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

\int 9 x^{25} d x = \frac{9 x ^{26}}{26}

\int 36 x^{23} d x = \frac{3 x ^{24}}{2}

\int 84 x^{21} d x = \frac{42 x ^{22}}{11}

\int 126 x^{19} d x = \frac{63 x ^{20}}{10}

\int 126 x^{17} d x = 7 x^{18}

\int 84 x^{15} d x = \frac{21 x ^{16}}{4}

\int 36 x^{13} d x = \frac{18 x ^{14}}{7}

\int 9 x^{11} d x = \frac{3 x ^{12}}{4}

\int x^{9} d x = \frac{x ^{10}}{10}

= \frac{x ^{28}}{28} + \frac{9 x ^{26}}{26} + \frac{3 x ^{24}}{2} + \frac{42 x ^{22}}{11} + \frac{63 x ^{20}}{10} + 7 x^{18} + \frac{21 x ^{16}}{4} + \frac{18 x ^{14}}{7} + \frac{3 x ^{12}}{4} + \frac{x ^{10}}{10}

定数を解答に追加する

= \frac{x ^{28}}{28} + \frac{9 x ^{26}}{26} + \frac{3 x ^{24}}{2} + \frac{42 x ^{22}}{11} + \frac{63 x ^{20}}{10} + 7 x^{18} + \frac{21 x ^{16}}{4} + \frac{18 x ^{14}}{7} + \frac{3 x ^{12}}{4} + \frac{x ^{10}}{10} + C