integral of 6x(-x^2+5)^7

解答

\int 6 x (- x^{2} + 5)^{7} d x

6 (- \frac{x ^{16}}{16} + \frac{5 x ^{14}}{2} - \frac{175 x ^{12}}{4} + \frac{875 x ^{10}}{2} - \frac{21875 x ^{8}}{8} + \frac{21875 x ^{6}}{2} - \frac{109375 x ^{4}}{4} + \frac{78125 x ^{2}}{2}) + C

求解步骤

\int 6 x (- x^{2} + 5)^{7} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int x (- x^{2} + 5)^{7} d x

乘开

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

= 6 \cdot \int - x^{15} + 35 x^{13} - 525 x^{11} + 4375 x^{9} - 21875 x^{7} + 65625 x^{5} - 109375 x^{3} + 78125 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

= 6 (- \int x^{15} d x + \int 35 x^{13} d x - \int 525 x^{11} d x + \int 4375 x^{9} d x - \int 21875 x^{7} d x + \int 65625 x^{5} d x - \int 109375 x^{3} d x + \int 78125 x d x)

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

\int 35 x^{13} d x = \frac{5 x ^{14}}{2}

\int 525 x^{11} d x = \frac{175 x ^{12}}{4}

\int 4375 x^{9} d x = \frac{875 x ^{10}}{2}

\int 21875 x^{7} d x = \frac{21875 x ^{8}}{8}

\int 65625 x^{5} d x = \frac{21875 x ^{6}}{2}

\int 109375 x^{3} d x = \frac{109375 x ^{4}}{4}

\int 78125 x d x = \frac{78125 x ^{2}}{2}

= 6 (- \frac{x ^{16}}{16} + \frac{5 x ^{14}}{2} - \frac{175 x ^{12}}{4} + \frac{875 x ^{10}}{2} - \frac{21875 x ^{8}}{8} + \frac{21875 x ^{6}}{2} - \frac{109375 x ^{4}}{4} + \frac{78125 x ^{2}}{2})

解答补常数

= 6 (- \frac{x ^{16}}{16} + \frac{5 x ^{14}}{2} - \frac{175 x ^{12}}{4} + \frac{875 x ^{10}}{2} - \frac{21875 x ^{8}}{8} + \frac{21875 x ^{6}}{2} - \frac{109375 x ^{4}}{4} + \frac{78125 x ^{2}}{2}) + C

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