integral of v(1-2/3 v^2)^6

解答

\int v (1 - \frac{2}{3} v^{2})^{6} d v

\frac{v ^{2}}{2} - v^{4} + \frac{10 v ^{6}}{9} - \frac{20 v ^{8}}{27} + \frac{8 v ^{10}}{27} - \frac{16 v ^{12}}{243} + \frac{32 v ^{14}}{5103} + C

求解步骤

\int v (1 - \frac{2}{3} v^{2})^{6} d v

乘开

v (1 - \frac{2}{3} v^{2})^{6} : v - 4 v^{3} + \frac{20 v ^{5}}{3} - \frac{160 v ^{7}}{27} + \frac{80 v ^{9}}{27} - \frac{64 v ^{11}}{81} + \frac{64 v ^{13}}{729}

= \int v - 4 v^{3} + \frac{20 v ^{5}}{3} - \frac{160 v ^{7}}{27} + \frac{80 v ^{9}}{27} - \frac{64 v ^{11}}{81} + \frac{64 v ^{13}}{729} d v

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 v d v - \int 4 v^{3} d v + \int \frac{20 v ^{5}}{3} d v - \int \frac{160 v ^{7}}{27} d v + \int \frac{80 v ^{9}}{27} d v - \int \frac{64 v ^{11}}{81} d v + \int \frac{64 v ^{13}}{729} d v

\int v d v = \frac{v ^{2}}{2}

\int 4 v^{3} d v = v^{4}

\int \frac{20 v ^{5}}{3} d v = \frac{10 v ^{6}}{9}

\int \frac{160 v ^{7}}{27} d v = \frac{20 v ^{8}}{27}

\int \frac{80 v ^{9}}{27} d v = \frac{8 v ^{10}}{27}

\int \frac{64 v ^{11}}{81} d v = \frac{16 v ^{12}}{243}

\int \frac{64 v ^{13}}{729} d v = \frac{32 v ^{14}}{5103}

= \frac{v ^{2}}{2} - v^{4} + \frac{10 v ^{6}}{9} - \frac{20 v ^{8}}{27} + \frac{8 v ^{10}}{27} - \frac{16 v ^{12}}{243} + \frac{32 v ^{14}}{5103}

解答补常数

= \frac{v ^{2}}{2} - v^{4} + \frac{10 v ^{6}}{9} - \frac{20 v ^{8}}{27} + \frac{8 v ^{10}}{27} - \frac{16 v ^{12}}{243} + \frac{32 v ^{14}}{5103} + C

\frac{\partial}{\partial x} (4 y cos (- 6 x))

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