integral of (x^6-2x^3)^5(x^5-x^2)

解答

\int (x^{6} - 2 x^{3})^{5} (x^{5} - x^{2}) d x

\frac{x ^{36}}{36} - \frac{x ^{33}}{3} + \frac{5 x ^{30}}{3} - \frac{40 x ^{27}}{9} + \frac{20 x ^{24}}{3} - \frac{16 x ^{21}}{3} + \frac{16 x ^{18}}{9} + C

求解步骤

\int (x^{6} - 2 x^{3})^{5} (x^{5} - x^{2}) d x

乘开

(x^{6} - 2 x^{3})^{5} (x^{5} - x^{2}) : x^{35} - 11 x^{32} + 50 x^{29} - 120 x^{26} + 160 x^{23} - 112 x^{20} + 32 x^{17}

= \int x^{35} - 11 x^{32} + 50 x^{29} - 120 x^{26} + 160 x^{23} - 112 x^{20} + 32 x^{17} 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^{35} d x - \int 11 x^{32} d x + \int 50 x^{29} d x - \int 120 x^{26} d x + \int 160 x^{23} d x - \int 112 x^{20} d x + \int 32 x^{17} d x

\int x^{35} d x = \frac{x ^{36}}{36}

\int 11 x^{32} d x = \frac{x ^{33}}{3}

\int 50 x^{29} d x = \frac{5 x ^{30}}{3}

\int 120 x^{26} d x = \frac{40 x ^{27}}{9}

\int 160 x^{23} d x = \frac{20 x ^{24}}{3}

\int 112 x^{20} d x = \frac{16 x ^{21}}{3}

\int 32 x^{17} d x = \frac{16 x ^{18}}{9}

= \frac{x ^{36}}{36} - \frac{x ^{33}}{3} + \frac{5 x ^{30}}{3} - \frac{40 x ^{27}}{9} + \frac{20 x ^{24}}{3} - \frac{16 x ^{21}}{3} + \frac{16 x ^{18}}{9}

解答补常数

= \frac{x ^{36}}{36} - \frac{x ^{33}}{3} + \frac{5 x ^{30}}{3} - \frac{40 x ^{27}}{9} + \frac{20 x ^{24}}{3} - \frac{16 x ^{21}}{3} + \frac{16 x ^{18}}{9} + C

d er i v a t i v e \frac{1}{3 ( 8 - x ^{3} ) ^{8}}

t an g e n t y = x^{2} - x, (4, 12)

\int \frac{1}{x ^{2} ( x + 1 ) ^{3}} d x

d er i v a t i v e f (x) = \frac{5 x}{x - 3}

n = 1 \sum \infty (2)^{n - 1}