integral of x(5x^2+1)^7

解答

\int x (5 x^{2} + 1)^{7} d x

\frac{78125 x ^{16}}{16} + \frac{15625 x ^{14}}{2} + \frac{21875 x ^{12}}{4} + \frac{4375 x ^{10}}{2} + \frac{4375 x ^{8}}{8} + \frac{175 x ^{6}}{2} + \frac{35 x ^{4}}{4} + \frac{x ^{2}}{2} + C

求解步骤

\int x (5 x^{2} + 1)^{7} d x

乘开

x (5 x^{2} + 1)^{7} : 78125 x^{15} + 109375 x^{13} + 65625 x^{11} + 21875 x^{9} + 4375 x^{7} + 525 x^{5} + 35 x^{3} + x

= \int 78125 x^{15} + 109375 x^{13} + 65625 x^{11} + 21875 x^{9} + 4375 x^{7} + 525 x^{5} + 35 x^{3} + 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 78125 x^{15} d x + \int 109375 x^{13} d x + \int 65625 x^{11} d x + \int 21875 x^{9} d x + \int 4375 x^{7} d x + \int 525 x^{5} d x + \int 35 x^{3} d x + \int x d x

\int 78125 x^{15} d x = \frac{78125 x ^{16}}{16}

\int 109375 x^{13} d x = \frac{15625 x ^{14}}{2}

\int 65625 x^{11} d x = \frac{21875 x ^{12}}{4}

\int 21875 x^{9} d x = \frac{4375 x ^{10}}{2}

\int 4375 x^{7} d x = \frac{4375 x ^{8}}{8}

\int 525 x^{5} d x = \frac{175 x ^{6}}{2}

\int 35 x^{3} d x = \frac{35 x ^{4}}{4}

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

= \frac{78125 x ^{16}}{16} + \frac{15625 x ^{14}}{2} + \frac{21875 x ^{12}}{4} + \frac{4375 x ^{10}}{2} + \frac{4375 x ^{8}}{8} + \frac{175 x ^{6}}{2} + \frac{35 x ^{4}}{4} + \frac{x ^{2}}{2}

解答补常数

= \frac{78125 x ^{16}}{16} + \frac{15625 x ^{14}}{2} + \frac{21875 x ^{12}}{4} + \frac{4375 x ^{10}}{2} + \frac{4375 x ^{8}}{8} + \frac{175 x ^{6}}{2} + \frac{35 x ^{4}}{4} + \frac{x ^{2}}{2} + C

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