integral of x^3(x+7)^9

解答

\int x^{3} (x + 7)^{9} d x

\frac{x ^{13}}{13} + \frac{21 x ^{12}}{4} + \frac{1764 x ^{11}}{11} + \frac{14406 x ^{10}}{5} + 33614 x^{9} + \frac{1058841 x ^{8}}{4} + 1411788 x^{7} + 4941258 x^{6} + \frac{51883209 x ^{5}}{5} + \frac{40353607 x ^{4}}{4} + C

求解步骤

\int x^{3} (x + 7)^{9} d x

乘开

x^{3} (x + 7)^{9} : x^{12} + 63 x^{11} + 1764 x^{10} + 28812 x^{9} + 302526 x^{8} + 2117682 x^{7} + 9882516 x^{6} + 29647548 x^{5} + 51883209 x^{4} + 40353607 x^{3}

= \int x^{12} + 63 x^{11} + 1764 x^{10} + 28812 x^{9} + 302526 x^{8} + 2117682 x^{7} + 9882516 x^{6} + 29647548 x^{5} + 51883209 x^{4} + 40353607 x^{3} 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^{12} d x + \int 63 x^{11} d x + \int 1764 x^{10} d x + \int 28812 x^{9} d x + \int 302526 x^{8} d x + \int 2117682 x^{7} d x + \int 9882516 x^{6} d x + \int 29647548 x^{5} d x + \int 51883209 x^{4} d x + \int 40353607 x^{3} d x

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

\int 63 x^{11} d x = \frac{21 x ^{12}}{4}

\int 1764 x^{10} d x = \frac{1764 x ^{11}}{11}

\int 28812 x^{9} d x = \frac{14406 x ^{10}}{5}

\int 302526 x^{8} d x = 33614 x^{9}

\int 2117682 x^{7} d x = \frac{1058841 x ^{8}}{4}

\int 9882516 x^{6} d x = 1411788 x^{7}

\int 29647548 x^{5} d x = 4941258 x^{6}

\int 51883209 x^{4} d x = \frac{51883209 x ^{5}}{5}

\int 40353607 x^{3} d x = \frac{40353607 x ^{4}}{4}

= \frac{x ^{13}}{13} + \frac{21 x ^{12}}{4} + \frac{1764 x ^{11}}{11} + \frac{14406 x ^{10}}{5} + 33614 x^{9} + \frac{1058841 x ^{8}}{4} + 1411788 x^{7} + 4941258 x^{6} + \frac{51883209 x ^{5}}{5} + \frac{40353607 x ^{4}}{4}

解答补常数

= \frac{x ^{13}}{13} + \frac{21 x ^{12}}{4} + \frac{1764 x ^{11}}{11} + \frac{14406 x ^{10}}{5} + 33614 x^{9} + \frac{1058841 x ^{8}}{4} + 1411788 x^{7} + 4941258 x^{6} + \frac{51883209 x ^{5}}{5} + \frac{40353607 x ^{4}}{4} + C