integral of (4x^2+8x+5)^6(x+1)

解答

\int (4 x^{2} + 8 x + 5)^{6} (x + 1) d x

\frac{2048 x ^{14}}{7} + 4096 x^{13} + 27136 x^{12} + 112640 x^{11} + 327040 x^{10} + 702208 x^{9} + 1149472 x^{8} + \frac{10198784 x ^{7}}{7} + 1436840 x^{6} + 1097200 x^{5} + 638750 x^{4} + 275000 x^{3} + \frac{165625 x ^{2}}{2} + 15625 x + C

求解步骤

\int (4 x^{2} + 8 x + 5)^{6} (x + 1) d x

乘开

(4 x^{2} + 8 x + 5)^{6} (x + 1) : 4096 x^{13} + 53248 x^{12} + 325632 x^{11} + 1239040 x^{10} + 3270400 x^{9} + 6319872 x^{8} + 9195776 x^{7} + 10198784 x^{6} + 8621040 x^{5} + 5486000 x^{4} + 2555000 x^{3} + 825000 x^{2} + 165625 x + 15625

= \int 4096 x^{13} + 53248 x^{12} + 325632 x^{11} + 1239040 x^{10} + 3270400 x^{9} + 6319872 x^{8} + 9195776 x^{7} + 10198784 x^{6} + 8621040 x^{5} + 5486000 x^{4} + 2555000 x^{3} + 825000 x^{2} + 165625 x + 15625 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 4096 x^{13} d x + \int 53248 x^{12} d x + \int 325632 x^{11} d x + \int 1239040 x^{10} d x + \int 3270400 x^{9} d x + \int 6319872 x^{8} d x + \int 9195776 x^{7} d x + \int 10198784 x^{6} d x + \int 8621040 x^{5} d x + \int 5486000 x^{4} d x + \int 2555000 x^{3} d x + \int 825000 x^{2} d x + \int 165625 x d x + \int 15625 d x

\int 4096 x^{13} d x = \frac{2048 x ^{14}}{7}

\int 53248 x^{12} d x = 4096 x^{13}

\int 325632 x^{11} d x = 27136 x^{12}

\int 1239040 x^{10} d x = 112640 x^{11}

\int 3270400 x^{9} d x = 327040 x^{10}

\int 6319872 x^{8} d x = 702208 x^{9}

\int 9195776 x^{7} d x = 1149472 x^{8}

\int 10198784 x^{6} d x = \frac{10198784 x ^{7}}{7}

\int 8621040 x^{5} d x = 1436840 x^{6}

\int 5486000 x^{4} d x = 1097200 x^{5}

\int 2555000 x^{3} d x = 638750 x^{4}

\int 825000 x^{2} d x = 275000 x^{3}

\int 165625 x d x = \frac{165625 x ^{2}}{2}

\int 15625 d x = 15625 x

= \frac{2048 x ^{14}}{7} + 4096 x^{13} + 27136 x^{12} + 112640 x^{11} + 327040 x^{10} + 702208 x^{9} + 1149472 x^{8} + \frac{10198784 x ^{7}}{7} + 1436840 x^{6} + 1097200 x^{5} + 638750 x^{4} + 275000 x^{3} + \frac{165625 x ^{2}}{2} + 15625 x

解答补常数

= \frac{2048 x ^{14}}{7} + 4096 x^{13} + 27136 x^{12} + 112640 x^{11} + 327040 x^{10} + 702208 x^{9} + 1149472 x^{8} + \frac{10198784 x ^{7}}{7} + 1436840 x^{6} + 1097200 x^{5} + 638750 x^{4} + 275000 x^{3} + \frac{165625 x ^{2}}{2} + 15625 x + C