integral of (3x^5-8)^96x^4

解答

\int (3 x^{5} - 8)^{9} 6 x^{4} d x

6 (\frac{19683 x ^{50}}{50} - \frac{52488 x ^{45}}{5} + \frac{629856 x ^{40}}{5} - \frac{4478976 x ^{35}}{5} + \frac{20901888 x ^{30}}{5} - \frac{334430208 x ^{25}}{25} + \frac{148635648 x ^{20}}{5} - \frac{226492416 x ^{15}}{5} + \frac{226492416 x ^{10}}{5} - \frac{134217728 x ^{5}}{5}) + C

求解步骤

\int (3 x^{5} - 8)^{9} \cdot 6 x^{4} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int (3 x^{5} - 8)^{9} x^{4} d x

乘开

(3 x^{5} - 8)^{9} x^{4} : 19683 x^{49} - 472392 x^{44} + 5038848 x^{39} - 31352832 x^{34} + 125411328 x^{29} - 334430208 x^{24} + 594542592 x^{19} - 679477248 x^{14} + 452984832 x^{9} - 134217728 x^{4}

= 6 \cdot \int 19683 x^{49} - 472392 x^{44} + 5038848 x^{39} - 31352832 x^{34} + 125411328 x^{29} - 334430208 x^{24} + 594542592 x^{19} - 679477248 x^{14} + 452984832 x^{9} - 134217728 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

= 6 (\int 19683 x^{49} d x - \int 472392 x^{44} d x + \int 5038848 x^{39} d x - \int 31352832 x^{34} d x + \int 125411328 x^{29} d x - \int 334430208 x^{24} d x + \int 594542592 x^{19} d x - \int 679477248 x^{14} d x + \int 452984832 x^{9} d x - \int 134217728 x^{4} d x)

\int 19683 x^{49} d x = \frac{19683 x ^{50}}{50}

\int 472392 x^{44} d x = \frac{52488 x ^{45}}{5}

\int 5038848 x^{39} d x = \frac{629856 x ^{40}}{5}

\int 31352832 x^{34} d x = \frac{4478976 x ^{35}}{5}

\int 125411328 x^{29} d x = \frac{20901888 x ^{30}}{5}

\int 334430208 x^{24} d x = \frac{334430208 x ^{25}}{25}

\int 594542592 x^{19} d x = \frac{148635648 x ^{20}}{5}

\int 679477248 x^{14} d x = \frac{226492416 x ^{15}}{5}

\int 452984832 x^{9} d x = \frac{226492416 x ^{10}}{5}

\int 134217728 x^{4} d x = \frac{134217728 x ^{5}}{5}

= 6 (\frac{19683 x ^{50}}{50} - \frac{52488 x ^{45}}{5} + \frac{629856 x ^{40}}{5} - \frac{4478976 x ^{35}}{5} + \frac{20901888 x ^{30}}{5} - \frac{334430208 x ^{25}}{25} + \frac{148635648 x ^{20}}{5} - \frac{226492416 x ^{15}}{5} + \frac{226492416 x ^{10}}{5} - \frac{134217728 x ^{5}}{5})

解答补常数

= 6 (\frac{19683 x ^{50}}{50} - \frac{52488 x ^{45}}{5} + \frac{629856 x ^{40}}{5} - \frac{4478976 x ^{35}}{5} + \frac{20901888 x ^{30}}{5} - \frac{334430208 x ^{25}}{25} + \frac{148635648 x ^{20}}{5} - \frac{226492416 x ^{15}}{5} + \frac{226492416 x ^{10}}{5} - \frac{134217728 x ^{5}}{5}) + C