integral of (2x-4)(x^2-4x+3)^5

解答

\int (2 x - 4) (x^{2} - 4 x + 3)^{5} d x

\frac{x ^{12}}{6} - 4 x^{11} + 43 x^{10} - \frac{820 x ^{9}}{3} + \frac{2285 x ^{8}}{2} - 3304 x^{7} + \frac{20318 x ^{6}}{3} - 9912 x^{5} + \frac{20565 x ^{4}}{2} - 7380 x^{3} + 3483 x^{2} - 972 x + C

求解步骤

\int (2 x - 4) (x^{2} - 4 x + 3)^{5} d x

乘开

(2 x - 4) (x^{2} - 4 x + 3)^{5} : 2 x^{11} - 44 x^{10} + 430 x^{9} - 2460 x^{8} + 9140 x^{7} - 23128 x^{6} + 40636 x^{5} - 49560 x^{4} + 41130 x^{3} - 22140 x^{2} + 6966 x - 972

= \int 2 x^{11} - 44 x^{10} + 430 x^{9} - 2460 x^{8} + 9140 x^{7} - 23128 x^{6} + 40636 x^{5} - 49560 x^{4} + 41130 x^{3} - 22140 x^{2} + 6966 x - 972 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 2 x^{11} d x - \int 44 x^{10} d x + \int 430 x^{9} d x - \int 2460 x^{8} d x + \int 9140 x^{7} d x - \int 23128 x^{6} d x + \int 40636 x^{5} d x - \int 49560 x^{4} d x + \int 41130 x^{3} d x - \int 22140 x^{2} d x + \int 6966 x d x - \int 972 d x

\int 2 x^{11} d x = \frac{x ^{12}}{6}

\int 44 x^{10} d x = 4 x^{11}

\int 430 x^{9} d x = 43 x^{10}

\int 2460 x^{8} d x = \frac{820 x ^{9}}{3}

\int 9140 x^{7} d x = \frac{2285 x ^{8}}{2}

\int 23128 x^{6} d x = 3304 x^{7}

\int 40636 x^{5} d x = \frac{20318 x ^{6}}{3}

\int 49560 x^{4} d x = 9912 x^{5}

\int 41130 x^{3} d x = \frac{20565 x ^{4}}{2}

\int 22140 x^{2} d x = 7380 x^{3}

\int 6966 x d x = 3483 x^{2}

\int 972 d x = 972 x

= \frac{x ^{12}}{6} - 4 x^{11} + 43 x^{10} - \frac{820 x ^{9}}{3} + \frac{2285 x ^{8}}{2} - 3304 x^{7} + \frac{20318 x ^{6}}{3} - 9912 x^{5} + \frac{20565 x ^{4}}{2} - 7380 x^{3} + 3483 x^{2} - 972 x

解答补常数

= \frac{x ^{12}}{6} - 4 x^{11} + 43 x^{10} - \frac{820 x ^{9}}{3} + \frac{2285 x ^{8}}{2} - 3304 x^{7} + \frac{20318 x ^{6}}{3} - 9912 x^{5} + \frac{20565 x ^{4}}{2} - 7380 x^{3} + 3483 x^{2} - 972 x + C

2 t y^{^{'}} + 4 y = 3

\int \frac{9}{x - 4} d x

\frac{d}{d x} (tan (4 x^{5}))

d er i v a t i v e y = \frac{t}{( t - 8 ) ^{2}}

2 y^{^{''}} + 18 y = 6 tan (3 t)