integral of x(x+9)^{10}

解

\int x (x + 9)^{10} d x

\frac{x ^{12}}{12} + \frac{90 x ^{11}}{11} + \frac{729 x ^{10}}{2} + 9720 x^{9} + \frac{688905 x ^{8}}{4} + 2125764 x^{7} + 18600435 x^{6} + 114791256 x^{5} + \frac{1937102445 x ^{4}}{4} + 1291401630 x^{3} + \frac{9 ^{10} x ^{2}}{2} + C

解答ステップ

\int x (x + 9)^{10} d x

拡張

x (x + 9)^{10} : x^{11} + 90 x^{10} + 3645 x^{9} + 87480 x^{8} + 1377810 x^{7} + 14880348 x^{6} + 111602610 x^{5} + 573956280 x^{4} + 1937102445 x^{3} + 3874204890 x^{2} + 9^{10} x

= \int x^{11} + 90 x^{10} + 3645 x^{9} + 87480 x^{8} + 1377810 x^{7} + 14880348 x^{6} + 111602610 x^{5} + 573956280 x^{4} + 1937102445 x^{3} + 3874204890 x^{2} + 9^{10} x 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 90 x^{10} d x + \int 3645 x^{9} d x + \int 87480 x^{8} d x + \int 1377810 x^{7} d x + \int 14880348 x^{6} d x + \int 111602610 x^{5} d x + \int 573956280 x^{4} d x + \int 1937102445 x^{3} d x + \int 3874204890 x^{2} d x + \int 9^{10} x d x

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

\int 90 x^{10} d x = \frac{90 x ^{11}}{11}

\int 3645 x^{9} d x = \frac{729 x ^{10}}{2}

\int 87480 x^{8} d x = 9720 x^{9}

\int 1377810 x^{7} d x = \frac{688905 x ^{8}}{4}

\int 14880348 x^{6} d x = 2125764 x^{7}

\int 111602610 x^{5} d x = 18600435 x^{6}

\int 573956280 x^{4} d x = 114791256 x^{5}

\int 1937102445 x^{3} d x = \frac{1937102445 x ^{4}}{4}

\int 3874204890 x^{2} d x = 1291401630 x^{3}

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

= \frac{x ^{12}}{12} + \frac{90 x ^{11}}{11} + \frac{729 x ^{10}}{2} + 9720 x^{9} + \frac{688905 x ^{8}}{4} + 2125764 x^{7} + 18600435 x^{6} + 114791256 x^{5} + \frac{1937102445 x ^{4}}{4} + 1291401630 x^{3} + \frac{9 ^{10} x ^{2}}{2}

定数を解答に追加する

= \frac{x ^{12}}{12} + \frac{90 x ^{11}}{11} + \frac{729 x ^{10}}{2} + 9720 x^{9} + \frac{688905 x ^{8}}{4} + 2125764 x^{7} + 18600435 x^{6} + 114791256 x^{5} + \frac{1937102445 x ^{4}}{4} + 1291401630 x^{3} + \frac{9 ^{10} x ^{2}}{2} + C