integral of x(x+4)^6

解答

\int x (x + 4)^{6} d x

\frac{x ^{8}}{8} + \frac{24 x ^{7}}{7} + 40 x^{6} + 256 x^{5} + 960 x^{4} + 2048 x^{3} + 2048 x^{2} + C

求解步骤

\int x (x + 4)^{6} d x

乘开

x (x + 4)^{6} : x^{7} + 24 x^{6} + 240 x^{5} + 1280 x^{4} + 3840 x^{3} + 6144 x^{2} + 4096 x

= \int x^{7} + 24 x^{6} + 240 x^{5} + 1280 x^{4} + 3840 x^{3} + 6144 x^{2} + 4096 x 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^{7} d x + \int 24 x^{6} d x + \int 240 x^{5} d x + \int 1280 x^{4} d x + \int 3840 x^{3} d x + \int 6144 x^{2} d x + \int 4096 x d x

\int x^{7} d x = \frac{x ^{8}}{8}

\int 24 x^{6} d x = \frac{24 x ^{7}}{7}

\int 240 x^{5} d x = 40 x^{6}

\int 1280 x^{4} d x = 256 x^{5}

\int 3840 x^{3} d x = 960 x^{4}

\int 6144 x^{2} d x = 2048 x^{3}

\int 4096 x d x = 2048 x^{2}

= \frac{x ^{8}}{8} + \frac{24 x ^{7}}{7} + 40 x^{6} + 256 x^{5} + 960 x^{4} + 2048 x^{3} + 2048 x^{2}

解答补常数

= \frac{x ^{8}}{8} + \frac{24 x ^{7}}{7} + 40 x^{6} + 256 x^{5} + 960 x^{4} + 2048 x^{3} + 2048 x^{2} + C

\int_{0}^{1} cos^{2} (x) d x

\frac{d}{d x} (ln (6 x + 1))

f (x) = x^{2} 2 x + 4

\int 3 tan (2 x) sec^{2} (2 x) d x

x^{2} \frac{d y}{d x} + x y = 3