integral of (7x^5+1)^9x^4

解答

\int (7 x^{5} + 1)^{9} x^{4} d x

\frac{40353607 x ^{50}}{50} + \frac{5764801 x ^{45}}{5} + \frac{7411887 x ^{40}}{10} + \frac{1411788 x ^{35}}{5} + \frac{352947 x ^{30}}{5} + \frac{302526 x ^{25}}{25} + \frac{7203 x ^{20}}{5} + \frac{588 x ^{15}}{5} + \frac{63 x ^{10}}{10} + \frac{x ^{5}}{5} + C

求解步骤

\int (7 x^{5} + 1)^{9} x^{4} d x

乘开

(7 x^{5} + 1)^{9} x^{4} : 40353607 x^{49} + 51883209 x^{44} + 29647548 x^{39} + 9882516 x^{34} + 2117682 x^{29} + 302526 x^{24} + 28812 x^{19} + 1764 x^{14} + 63 x^{9} + x^{4}

= \int 40353607 x^{49} + 51883209 x^{44} + 29647548 x^{39} + 9882516 x^{34} + 2117682 x^{29} + 302526 x^{24} + 28812 x^{19} + 1764 x^{14} + 63 x^{9} + x^{4} 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 40353607 x^{49} d x + \int 51883209 x^{44} d x + \int 29647548 x^{39} d x + \int 9882516 x^{34} d x + \int 2117682 x^{29} d x + \int 302526 x^{24} d x + \int 28812 x^{19} d x + \int 1764 x^{14} d x + \int 63 x^{9} d x + \int x^{4} d x

\int 40353607 x^{49} d x = \frac{40353607 x ^{50}}{50}

\int 51883209 x^{44} d x = \frac{5764801 x ^{45}}{5}

\int 29647548 x^{39} d x = \frac{7411887 x ^{40}}{10}

\int 9882516 x^{34} d x = \frac{1411788 x ^{35}}{5}

\int 2117682 x^{29} d x = \frac{352947 x ^{30}}{5}

\int 302526 x^{24} d x = \frac{302526 x ^{25}}{25}

\int 28812 x^{19} d x = \frac{7203 x ^{20}}{5}

\int 1764 x^{14} d x = \frac{588 x ^{15}}{5}

\int 63 x^{9} d x = \frac{63 x ^{10}}{10}

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

= \frac{40353607 x ^{50}}{50} + \frac{5764801 x ^{45}}{5} + \frac{7411887 x ^{40}}{10} + \frac{1411788 x ^{35}}{5} + \frac{352947 x ^{30}}{5} + \frac{302526 x ^{25}}{25} + \frac{7203 x ^{20}}{5} + \frac{588 x ^{15}}{5} + \frac{63 x ^{10}}{10} + \frac{x ^{5}}{5}

解答补常数

= \frac{40353607 x ^{50}}{50} + \frac{5764801 x ^{45}}{5} + \frac{7411887 x ^{40}}{10} + \frac{1411788 x ^{35}}{5} + \frac{352947 x ^{30}}{5} + \frac{302526 x ^{25}}{25} + \frac{7203 x ^{20}}{5} + \frac{588 x ^{15}}{5} + \frac{63 x ^{10}}{10} + \frac{x ^{5}}{5} + C

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