integral of (x^{20}-5x^4)^3(x^{19}-x^3)

解答

\int (x^{20} - 5 x^{4})^{3} (x^{19} - x^{3}) d x

\frac{x ^{80}}{80} - \frac{x ^{64}}{4} + \frac{15 x ^{48}}{8} - \frac{25 x ^{32}}{4} + \frac{125 x ^{16}}{16} + C

求解步骤

\int (x^{20} - 5 x^{4})^{3} (x^{19} - x^{3}) d x

乘开

(x^{20} - 5 x^{4})^{3} (x^{19} - x^{3}) : x^{79} - 16 x^{63} + 90 x^{47} - 200 x^{31} + 125 x^{15}

= \int x^{79} - 16 x^{63} + 90 x^{47} - 200 x^{31} + 125 x^{15} 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^{79} d x - \int 16 x^{63} d x + \int 90 x^{47} d x - \int 200 x^{31} d x + \int 125 x^{15} d x

\int x^{79} d x = \frac{x ^{80}}{80}

\int 16 x^{63} d x = \frac{x ^{64}}{4}

\int 90 x^{47} d x = \frac{15 x ^{48}}{8}

\int 200 x^{31} d x = \frac{25 x ^{32}}{4}

\int 125 x^{15} d x = \frac{125 x ^{16}}{16}

= \frac{x ^{80}}{80} - \frac{x ^{64}}{4} + \frac{15 x ^{48}}{8} - \frac{25 x ^{32}}{4} + \frac{125 x ^{16}}{16}

解答补常数

= \frac{x ^{80}}{80} - \frac{x ^{64}}{4} + \frac{15 x ^{48}}{8} - \frac{25 x ^{32}}{4} + \frac{125 x ^{16}}{16} + C