integral of (4x^2-16x+7)^4(x-2)

解答

\int (4 x^{2} - 16 x + 7)^{4} (x - 2) d x

\frac{128 x ^{10}}{5} - 512 x^{9} + 4320 x^{8} - 19968 x^{7} + 55056 x^{6} - \frac{464832 x ^{5}}{5} + 96348 x^{4} - 61152 x^{3} + \frac{46305 x ^{2}}{2} - 4802 x + C

求解步骤

\int (4 x^{2} - 16 x + 7)^{4} (x - 2) d x

乘开

(4 x^{2} - 16 x + 7)^{4} (x - 2) : 256 x^{9} - 4608 x^{8} + 34560 x^{7} - 139776 x^{6} + 330336 x^{5} - 464832 x^{4} + 385392 x^{3} - 183456 x^{2} + 46305 x - 4802

= \int 256 x^{9} - 4608 x^{8} + 34560 x^{7} - 139776 x^{6} + 330336 x^{5} - 464832 x^{4} + 385392 x^{3} - 183456 x^{2} + 46305 x - 4802 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 256 x^{9} d x - \int 4608 x^{8} d x + \int 34560 x^{7} d x - \int 139776 x^{6} d x + \int 330336 x^{5} d x - \int 464832 x^{4} d x + \int 385392 x^{3} d x - \int 183456 x^{2} d x + \int 46305 x d x - \int 4802 d x

\int 256 x^{9} d x = \frac{128 x ^{10}}{5}

\int 4608 x^{8} d x = 512 x^{9}

\int 34560 x^{7} d x = 4320 x^{8}

\int 139776 x^{6} d x = 19968 x^{7}

\int 330336 x^{5} d x = 55056 x^{6}

\int 464832 x^{4} d x = \frac{464832 x ^{5}}{5}

\int 385392 x^{3} d x = 96348 x^{4}

\int 183456 x^{2} d x = 61152 x^{3}

\int 46305 x d x = \frac{46305 x ^{2}}{2}

\int 4802 d x = 4802 x

= \frac{128 x ^{10}}{5} - 512 x^{9} + 4320 x^{8} - 19968 x^{7} + 55056 x^{6} - \frac{464832 x ^{5}}{5} + 96348 x^{4} - 61152 x^{3} + \frac{46305 x ^{2}}{2} - 4802 x

解答补常数

= \frac{128 x ^{10}}{5} - 512 x^{9} + 4320 x^{8} - 19968 x^{7} + 55056 x^{6} - \frac{464832 x ^{5}}{5} + 96348 x^{4} - 61152 x^{3} + \frac{46305 x ^{2}}{2} - 4802 x + C

\int \frac{1}{81 + x ^{2}} d x

x \to 0 lim (0.1 ln (0.1))

\int sin^{2} (x) (cos (x)) d x

\frac{\partial}{\partial x} (x \cdot ln (t))

\frac{d v}{d t} = 9.81 - \frac{0.5}{v ^{- 1}}