integral of 24x^2(6-5x^3)^7

解答

\int 24 x^{2} (6 - 5 x^{3})^{7} d x

2239488 x^{3} - 6531840 x^{6} + 10886400 x^{9} - 11340000 x^{12} + 7560000 x^{15} - 3150000 x^{18} + 750000 x^{21} - 78125 x^{24} + C

求解步骤

\int 24 x^{2} (6 - 5 x^{3})^{7} d x

提出常数:

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

= 24 \cdot \int x^{2} (6 - 5 x^{3})^{7} d x

乘开

x^{2} (6 - 5 x^{3})^{7} : 279936 x^{2} - 1632960 x^{5} + 4082400 x^{8} - 5670000 x^{11} + 4725000 x^{14} - 2362500 x^{17} + 656250 x^{20} - 78125 x^{23}

= 24 \cdot \int 279936 x^{2} - 1632960 x^{5} + 4082400 x^{8} - 5670000 x^{11} + 4725000 x^{14} - 2362500 x^{17} + 656250 x^{20} - 78125 x^{23} 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

= 24 (\int 279936 x^{2} d x - \int 1632960 x^{5} d x + \int 4082400 x^{8} d x - \int 5670000 x^{11} d x + \int 4725000 x^{14} d x - \int 2362500 x^{17} d x + \int 656250 x^{20} d x - \int 78125 x^{23} d x)

\int 279936 x^{2} d x = 93312 x^{3}

\int 1632960 x^{5} d x = 272160 x^{6}

\int 4082400 x^{8} d x = 453600 x^{9}

\int 5670000 x^{11} d x = 472500 x^{12}

\int 4725000 x^{14} d x = 315000 x^{15}

\int 2362500 x^{17} d x = 131250 x^{18}

\int 656250 x^{20} d x = 31250 x^{21}

\int 78125 x^{23} d x = \frac{78125 x ^{24}}{24}

= 24 (93312 x^{3} - 272160 x^{6} + 453600 x^{9} - 472500 x^{12} + 315000 x^{15} - 131250 x^{18} + 31250 x^{21} - \frac{78125 x ^{24}}{24})

化简

24 (93312 x^{3} - 272160 x^{6} + 453600 x^{9} - 472500 x^{12} + 315000 x^{15} - 131250 x^{18} + 31250 x^{21} - \frac{78125 x ^{24}}{24}) : 2239488 x^{3} - 6531840 x^{6} + 10886400 x^{9} - 11340000 x^{12} + 7560000 x^{15} - 3150000 x^{18} + 750000 x^{21} - 78125 x^{24}

= 2239488 x^{3} - 6531840 x^{6} + 10886400 x^{9} - 11340000 x^{12} + 7560000 x^{15} - 3150000 x^{18} + 750000 x^{21} - 78125 x^{24}

解答补常数

= 2239488 x^{3} - 6531840 x^{6} + 10886400 x^{9} - 11340000 x^{12} + 7560000 x^{15} - 3150000 x^{18} + 750000 x^{21} - 78125 x^{24} + C

t an g e n t f (x) = \frac{x}{9 + x ^{2}}, at x = 0

y^{^{''}} + 2 y^{^{'}} + 2 y = sin (5 t)

x^{2} \frac{d y}{d x} - 2 x y = 3 y^{4}, y (1) = \frac{1}{2}

\int ln (8 x) d x

d er i v a t i v e sin (5 x - 3)