integral of x^3(x^2+1)^{11}

解

\int x^{3} (x^{2} + 1)^{11} d x

\frac{x ^{26}}{26} + \frac{11 x ^{24}}{24} + \frac{5 x ^{22}}{2} + \frac{33 x ^{20}}{4} + \frac{55 x ^{18}}{3} + \frac{231 x ^{16}}{8} + 33 x^{14} + \frac{55 x ^{12}}{2} + \frac{33 x ^{10}}{2} + \frac{55 x ^{8}}{8} + \frac{11 x ^{6}}{6} + \frac{x ^{4}}{4} + C

解答ステップ

\int x^{3} (x^{2} + 1)^{11} d x

拡張

x^{3} (x^{2} + 1)^{11} : x^{25} + 11 x^{23} + 55 x^{21} + 165 x^{19} + 330 x^{17} + 462 x^{15} + 462 x^{13} + 330 x^{11} + 165 x^{9} + 55 x^{7} + 11 x^{5} + x^{3}

= \int x^{25} + 11 x^{23} + 55 x^{21} + 165 x^{19} + 330 x^{17} + 462 x^{15} + 462 x^{13} + 330 x^{11} + 165 x^{9} + 55 x^{7} + 11 x^{5} + x^{3} 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^{25} d x + \int 11 x^{23} d x + \int 55 x^{21} d x + \int 165 x^{19} d x + \int 330 x^{17} d x + \int 462 x^{15} d x + \int 462 x^{13} d x + \int 330 x^{11} d x + \int 165 x^{9} d x + \int 55 x^{7} d x + \int 11 x^{5} d x + \int x^{3} d x

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

\int 11 x^{23} d x = \frac{11 x ^{24}}{24}

\int 55 x^{21} d x = \frac{5 x ^{22}}{2}

\int 165 x^{19} d x = \frac{33 x ^{20}}{4}

\int 330 x^{17} d x = \frac{55 x ^{18}}{3}

\int 462 x^{15} d x = \frac{231 x ^{16}}{8}

\int 462 x^{13} d x = 33 x^{14}

\int 330 x^{11} d x = \frac{55 x ^{12}}{2}

\int 165 x^{9} d x = \frac{33 x ^{10}}{2}

\int 55 x^{7} d x = \frac{55 x ^{8}}{8}

\int 11 x^{5} d x = \frac{11 x ^{6}}{6}

\int x^{3} d x = \frac{x ^{4}}{4}

= \frac{x ^{26}}{26} + \frac{11 x ^{24}}{24} + \frac{5 x ^{22}}{2} + \frac{33 x ^{20}}{4} + \frac{55 x ^{18}}{3} + \frac{231 x ^{16}}{8} + 33 x^{14} + \frac{55 x ^{12}}{2} + \frac{33 x ^{10}}{2} + \frac{55 x ^{8}}{8} + \frac{11 x ^{6}}{6} + \frac{x ^{4}}{4}

定数を解答に追加する

= \frac{x ^{26}}{26} + \frac{11 x ^{24}}{24} + \frac{5 x ^{22}}{2} + \frac{33 x ^{20}}{4} + \frac{55 x ^{18}}{3} + \frac{231 x ^{16}}{8} + 33 x^{14} + \frac{55 x ^{12}}{2} + \frac{33 x ^{10}}{2} + \frac{55 x ^{8}}{8} + \frac{11 x ^{6}}{6} + \frac{x ^{4}}{4} + C