integral of (18x^2+3)(6x^3+3x)^6

解答

\int (18 x^{2} + 3) (6 x^{3} + 3 x)^{6} d x

\frac{279936 x ^{21}}{7} + 139968 x^{19} + 209952 x^{17} + 174960 x^{15} + 87480 x^{13} + 26244 x^{11} + 4374 x^{9} + \frac{2187 x ^{7}}{7} + C

求解步骤

\int (18 x^{2} + 3) (6 x^{3} + 3 x)^{6} d x

乘开

(18 x^{2} + 3) (6 x^{3} + 3 x)^{6} : 839808 x^{20} + 2659392 x^{18} + 3569184 x^{16} + 2624400 x^{14} + 1137240 x^{12} + 288684 x^{10} + 39366 x^{8} + 2187 x^{6}

= \int 839808 x^{20} + 2659392 x^{18} + 3569184 x^{16} + 2624400 x^{14} + 1137240 x^{12} + 288684 x^{10} + 39366 x^{8} + 2187 x^{6} 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 839808 x^{20} d x + \int 2659392 x^{18} d x + \int 3569184 x^{16} d x + \int 2624400 x^{14} d x + \int 1137240 x^{12} d x + \int 288684 x^{10} d x + \int 39366 x^{8} d x + \int 2187 x^{6} d x

\int 839808 x^{20} d x = \frac{279936 x ^{21}}{7}

\int 2659392 x^{18} d x = 139968 x^{19}

\int 3569184 x^{16} d x = 209952 x^{17}

\int 2624400 x^{14} d x = 174960 x^{15}

\int 1137240 x^{12} d x = 87480 x^{13}

\int 288684 x^{10} d x = 26244 x^{11}

\int 39366 x^{8} d x = 4374 x^{9}

\int 2187 x^{6} d x = \frac{2187 x ^{7}}{7}

= \frac{279936 x ^{21}}{7} + 139968 x^{19} + 209952 x^{17} + 174960 x^{15} + 87480 x^{13} + 26244 x^{11} + 4374 x^{9} + \frac{2187 x ^{7}}{7}

解答补常数

= \frac{279936 x ^{21}}{7} + 139968 x^{19} + 209952 x^{17} + 174960 x^{15} + 87480 x^{13} + 26244 x^{11} + 4374 x^{9} + \frac{2187 x ^{7}}{7} + C

\int \frac{2 x + 1}{( x + 1 ) ^{3} \cdot ( x ^{2} + 4 ) ^{2}} d x

s l o p e f (x) = 4^{x}

y = x^{4 x}

\int x tanh^{2} (x) d x

d er i v a t i v e e^{- \frac{1}{x}} + 4 x