integral of (x+2)^{15}

解答

\int (x + 2)^{15} d x

\frac{x ^{16}}{16} + 2 x^{15} + 30 x^{14} + 280 x^{13} + 1820 x^{12} + 8736 x^{11} + 32032 x^{10} + 91520 x^{9} + 205920 x^{8} + 366080 x^{7} + 512512 x^{6} + 559104 x^{5} + 465920 x^{4} + 286720 x^{3} + 122880 x^{2} + 32768 x + C

求解步骤

\int (x + 2)^{15} d x

乘开

(x + 2)^{15} : x^{15} + 30 x^{14} + 420 x^{13} + 3640 x^{12} + 21840 x^{11} + 96096 x^{10} + 320320 x^{9} + 823680 x^{8} + 1647360 x^{7} + 2562560 x^{6} + 3075072 x^{5} + 2795520 x^{4} + 1863680 x^{3} + 860160 x^{2} + 245760 x + 32768

= \int x^{15} + 30 x^{14} + 420 x^{13} + 3640 x^{12} + 21840 x^{11} + 96096 x^{10} + 320320 x^{9} + 823680 x^{8} + 1647360 x^{7} + 2562560 x^{6} + 3075072 x^{5} + 2795520 x^{4} + 1863680 x^{3} + 860160 x^{2} + 245760 x + 32768 d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{15} d x + \int 30 x^{14} d x + \int 420 x^{13} d x + \int 3640 x^{12} d x + \int 21840 x^{11} d x + \int 96096 x^{10} d x + \int 320320 x^{9} d x + \int 823680 x^{8} d x + \int 1647360 x^{7} d x + \int 2562560 x^{6} d x + \int 3075072 x^{5} d x + \int 2795520 x^{4} d x + \int 1863680 x^{3} d x + \int 860160 x^{2} d x + \int 245760 x d x + \int 32768 d x

\int x^{15} d x = \frac{x ^{16}}{16}

\int 30 x^{14} d x = 2 x^{15}

\int 420 x^{13} d x = 30 x^{14}

\int 3640 x^{12} d x = 280 x^{13}

\int 21840 x^{11} d x = 1820 x^{12}

\int 96096 x^{10} d x = 8736 x^{11}

\int 320320 x^{9} d x = 32032 x^{10}

\int 823680 x^{8} d x = 91520 x^{9}

\int 1647360 x^{7} d x = 205920 x^{8}

\int 2562560 x^{6} d x = 366080 x^{7}

\int 3075072 x^{5} d x = 512512 x^{6}

\int 2795520 x^{4} d x = 559104 x^{5}

\int 1863680 x^{3} d x = 465920 x^{4}

\int 860160 x^{2} d x = 286720 x^{3}

\int 245760 x d x = 122880 x^{2}

\int 32768 d x = 32768 x

= \frac{x ^{16}}{16} + 2 x^{15} + 30 x^{14} + 280 x^{13} + 1820 x^{12} + 8736 x^{11} + 32032 x^{10} + 91520 x^{9} + 205920 x^{8} + 366080 x^{7} + 512512 x^{6} + 559104 x^{5} + 465920 x^{4} + 286720 x^{3} + 122880 x^{2} + 32768 x

解答补常数

= \frac{x ^{16}}{16} + 2 x^{15} + 30 x^{14} + 280 x^{13} + 1820 x^{12} + 8736 x^{11} + 32032 x^{10} + 91520 x^{9} + 205920 x^{8} + 366080 x^{7} + 512512 x^{6} + 559104 x^{5} + 465920 x^{4} + 286720 x^{3} + 122880 x^{2} + 32768 x + C