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

解

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

\frac{x ^{13}}{13} + \frac{11 x ^{12}}{4} + 45 x^{11} + \frac{891 x ^{10}}{2} + 2970 x^{9} + \frac{56133 x ^{8}}{4} + 48114 x^{7} + 120285 x^{6} + 216513 x^{5} + \frac{1082565 x ^{4}}{4} + 216513 x^{3} + \frac{177147 x ^{2}}{2} + C

解答ステップ

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

拡張

x (x + 3)^{11} : x^{12} + 33 x^{11} + 495 x^{10} + 4455 x^{9} + 26730 x^{8} + 112266 x^{7} + 336798 x^{6} + 721710 x^{5} + 1082565 x^{4} + 1082565 x^{3} + 649539 x^{2} + 177147 x

= \int x^{12} + 33 x^{11} + 495 x^{10} + 4455 x^{9} + 26730 x^{8} + 112266 x^{7} + 336798 x^{6} + 721710 x^{5} + 1082565 x^{4} + 1082565 x^{3} + 649539 x^{2} + 177147 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^{12} d x + \int 33 x^{11} d x + \int 495 x^{10} d x + \int 4455 x^{9} d x + \int 26730 x^{8} d x + \int 112266 x^{7} d x + \int 336798 x^{6} d x + \int 721710 x^{5} d x + \int 1082565 x^{4} d x + \int 1082565 x^{3} d x + \int 649539 x^{2} d x + \int 177147 x d x

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

\int 33 x^{11} d x = \frac{11 x ^{12}}{4}

\int 495 x^{10} d x = 45 x^{11}

\int 4455 x^{9} d x = \frac{891 x ^{10}}{2}

\int 26730 x^{8} d x = 2970 x^{9}

\int 112266 x^{7} d x = \frac{56133 x ^{8}}{4}

\int 336798 x^{6} d x = 48114 x^{7}

\int 721710 x^{5} d x = 120285 x^{6}

\int 1082565 x^{4} d x = 216513 x^{5}

\int 1082565 x^{3} d x = \frac{1082565 x ^{4}}{4}

\int 649539 x^{2} d x = 216513 x^{3}

\int 177147 x d x = \frac{177147 x ^{2}}{2}

= \frac{x ^{13}}{13} + \frac{11 x ^{12}}{4} + 45 x^{11} + \frac{891 x ^{10}}{2} + 2970 x^{9} + \frac{56133 x ^{8}}{4} + 48114 x^{7} + 120285 x^{6} + 216513 x^{5} + \frac{1082565 x ^{4}}{4} + 216513 x^{3} + \frac{177147 x ^{2}}{2}

定数を解答に追加する

= \frac{x ^{13}}{13} + \frac{11 x ^{12}}{4} + 45 x^{11} + \frac{891 x ^{10}}{2} + 2970 x^{9} + \frac{56133 x ^{8}}{4} + 48114 x^{7} + 120285 x^{6} + 216513 x^{5} + \frac{1082565 x ^{4}}{4} + 216513 x^{3} + \frac{177147 x ^{2}}{2} + C