integral of x(x-7)^5

解答

\int x (x - 7)^{5} d x

\frac{x ^{7}}{7} - \frac{35 x ^{6}}{6} + 98 x^{5} - \frac{1715 x ^{4}}{2} + \frac{12005 x ^{3}}{3} - \frac{16807 x ^{2}}{2} + C

求解步骤

\int x (x - 7)^{5} d x

乘开

x (x - 7)^{5} : x^{6} - 35 x^{5} + 490 x^{4} - 3430 x^{3} + 12005 x^{2} - 16807 x

= \int x^{6} - 35 x^{5} + 490 x^{4} - 3430 x^{3} + 12005 x^{2} - 16807 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^{6} d x - \int 35 x^{5} d x + \int 490 x^{4} d x - \int 3430 x^{3} d x + \int 12005 x^{2} d x - \int 16807 x d x

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

\int 35 x^{5} d x = \frac{35 x ^{6}}{6}

\int 490 x^{4} d x = 98 x^{5}

\int 3430 x^{3} d x = \frac{1715 x ^{4}}{2}

\int 12005 x^{2} d x = \frac{12005 x ^{3}}{3}

\int 16807 x d x = \frac{16807 x ^{2}}{2}

= \frac{x ^{7}}{7} - \frac{35 x ^{6}}{6} + 98 x^{5} - \frac{1715 x ^{4}}{2} + \frac{12005 x ^{3}}{3} - \frac{16807 x ^{2}}{2}

解答补常数

= \frac{x ^{7}}{7} - \frac{35 x ^{6}}{6} + 98 x^{5} - \frac{1715 x ^{4}}{2} + \frac{12005 x ^{3}}{3} - \frac{16807 x ^{2}}{2} + C

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