integral of (x+1)^{10}(x+5)

解

\int (x + 1)^{10} (x + 5) d x

\frac{x ^{12}}{12} + \frac{15 x ^{11}}{11} + \frac{19 x ^{10}}{2} + \frac{115 x ^{9}}{3} + \frac{405 x ^{8}}{4} + 186 x^{7} + 245 x^{6} + 234 x^{5} + \frac{645 x ^{4}}{4} + \frac{235 x ^{3}}{3} + \frac{51 x ^{2}}{2} + 5 x + C

解答ステップ

\int (x + 1)^{10} (x + 5) d x

拡張

(x + 1)^{10} (x + 5) : x^{11} + 15 x^{10} + 95 x^{9} + 345 x^{8} + 810 x^{7} + 1302 x^{6} + 1470 x^{5} + 1170 x^{4} + 645 x^{3} + 235 x^{2} + 51 x + 5

= \int x^{11} + 15 x^{10} + 95 x^{9} + 345 x^{8} + 810 x^{7} + 1302 x^{6} + 1470 x^{5} + 1170 x^{4} + 645 x^{3} + 235 x^{2} + 51 x + 5 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^{11} d x + \int 15 x^{10} d x + \int 95 x^{9} d x + \int 345 x^{8} d x + \int 810 x^{7} d x + \int 1302 x^{6} d x + \int 1470 x^{5} d x + \int 1170 x^{4} d x + \int 645 x^{3} d x + \int 235 x^{2} d x + \int 51 x d x + \int 5 d x

\int x^{11} d x = \frac{x ^{12}}{12}

\int 15 x^{10} d x = \frac{15 x ^{11}}{11}

\int 95 x^{9} d x = \frac{19 x ^{10}}{2}

\int 345 x^{8} d x = \frac{115 x ^{9}}{3}

\int 810 x^{7} d x = \frac{405 x ^{8}}{4}

\int 1302 x^{6} d x = 186 x^{7}

\int 1470 x^{5} d x = 245 x^{6}

\int 1170 x^{4} d x = 234 x^{5}

\int 645 x^{3} d x = \frac{645 x ^{4}}{4}

\int 235 x^{2} d x = \frac{235 x ^{3}}{3}

\int 51 x d x = \frac{51 x ^{2}}{2}

\int 5 d x = 5 x

= \frac{x ^{12}}{12} + \frac{15 x ^{11}}{11} + \frac{19 x ^{10}}{2} + \frac{115 x ^{9}}{3} + \frac{405 x ^{8}}{4} + 186 x^{7} + 245 x^{6} + 234 x^{5} + \frac{645 x ^{4}}{4} + \frac{235 x ^{3}}{3} + \frac{51 x ^{2}}{2} + 5 x

定数を解答に追加する

= \frac{x ^{12}}{12} + \frac{15 x ^{11}}{11} + \frac{19 x ^{10}}{2} + \frac{115 x ^{9}}{3} + \frac{405 x ^{8}}{4} + 186 x^{7} + 245 x^{6} + 234 x^{5} + \frac{645 x ^{4}}{4} + \frac{235 x ^{3}}{3} + \frac{51 x ^{2}}{2} + 5 x + C