integral of (x^2+1)^{10}

解

\int (x^{2} + 1)^{10} d x

\frac{x ^{21}}{21} + \frac{10 x ^{19}}{19} + \frac{45 x ^{17}}{17} + 8 x^{15} + \frac{210 x ^{13}}{13} + \frac{252 x ^{11}}{11} + \frac{70 x ^{9}}{3} + \frac{120 x ^{7}}{7} + 9 x^{5} + \frac{10 x ^{3}}{3} + x + C

解答ステップ

\int (x^{2} + 1)^{10} d x

拡張

(x^{2} + 1)^{10} : x^{20} + 10 x^{18} + 45 x^{16} + 120 x^{14} + 210 x^{12} + 252 x^{10} + 210 x^{8} + 120 x^{6} + 45 x^{4} + 10 x^{2} + 1

= \int x^{20} + 10 x^{18} + 45 x^{16} + 120 x^{14} + 210 x^{12} + 252 x^{10} + 210 x^{8} + 120 x^{6} + 45 x^{4} + 10 x^{2} + 1 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^{20} d x + \int 10 x^{18} d x + \int 45 x^{16} d x + \int 120 x^{14} d x + \int 210 x^{12} d x + \int 252 x^{10} d x + \int 210 x^{8} d x + \int 120 x^{6} d x + \int 45 x^{4} d x + \int 10 x^{2} d x + \int 1 d x

\int x^{20} d x = \frac{x ^{21}}{21}

\int 10 x^{18} d x = \frac{10 x ^{19}}{19}

\int 45 x^{16} d x = \frac{45 x ^{17}}{17}

\int 120 x^{14} d x = 8 x^{15}

\int 210 x^{12} d x = \frac{210 x ^{13}}{13}

\int 252 x^{10} d x = \frac{252 x ^{11}}{11}

\int 210 x^{8} d x = \frac{70 x ^{9}}{3}

\int 120 x^{6} d x = \frac{120 x ^{7}}{7}

\int 45 x^{4} d x = 9 x^{5}

\int 10 x^{2} d x = \frac{10 x ^{3}}{3}

\int 1 d x = x

= \frac{x ^{21}}{21} + \frac{10 x ^{19}}{19} + \frac{45 x ^{17}}{17} + 8 x^{15} + \frac{210 x ^{13}}{13} + \frac{252 x ^{11}}{11} + \frac{70 x ^{9}}{3} + \frac{120 x ^{7}}{7} + 9 x^{5} + \frac{10 x ^{3}}{3} + x

定数を解答に追加する

= \frac{x ^{21}}{21} + \frac{10 x ^{19}}{19} + \frac{45 x ^{17}}{17} + 8 x^{15} + \frac{210 x ^{13}}{13} + \frac{252 x ^{11}}{11} + \frac{70 x ^{9}}{3} + \frac{120 x ^{7}}{7} + 9 x^{5} + \frac{10 x ^{3}}{3} + x + C