integral of (x^5-3)^6

解答

\int (x^{5} - 3)^{6} d x

\frac{x ^{31}}{31} - \frac{9 x ^{26}}{13} + \frac{45 x ^{21}}{7} - \frac{135 x ^{16}}{4} + \frac{1215 x ^{11}}{11} - 243 x^{6} + 729 x + C

求解步骤

\int (x^{5} - 3)^{6} d x

乘开

(x^{5} - 3)^{6} : x^{30} - 18 x^{25} + 135 x^{20} - 540 x^{15} + 1215 x^{10} - 1458 x^{5} + 729

= \int x^{30} - 18 x^{25} + 135 x^{20} - 540 x^{15} + 1215 x^{10} - 1458 x^{5} + 729 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^{30} d x - \int 18 x^{25} d x + \int 135 x^{20} d x - \int 540 x^{15} d x + \int 1215 x^{10} d x - \int 1458 x^{5} d x + \int 729 d x

\int x^{30} d x = \frac{x ^{31}}{31}

\int 18 x^{25} d x = \frac{9 x ^{26}}{13}

\int 135 x^{20} d x = \frac{45 x ^{21}}{7}

\int 540 x^{15} d x = \frac{135 x ^{16}}{4}

\int 1215 x^{10} d x = \frac{1215 x ^{11}}{11}

\int 1458 x^{5} d x = 243 x^{6}

\int 729 d x = 729 x

= \frac{x ^{31}}{31} - \frac{9 x ^{26}}{13} + \frac{45 x ^{21}}{7} - \frac{135 x ^{16}}{4} + \frac{1215 x ^{11}}{11} - 243 x^{6} + 729 x

解答补常数

= \frac{x ^{31}}{31} - \frac{9 x ^{26}}{13} + \frac{45 x ^{21}}{7} - \frac{135 x ^{16}}{4} + \frac{1215 x ^{11}}{11} - 243 x^{6} + 729 x + C

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