integral of (x^2+3x^{13})^5

解答

\int (x^{2} + 3 x^{13})^{5} d x

\frac{x ^{11}}{11} + \frac{15 x ^{22}}{22} + \frac{30 x ^{33}}{11} + \frac{135 x ^{44}}{22} + \frac{81 x ^{55}}{11} + \frac{81 x ^{66}}{22} + C

求解步骤

\int (x^{2} + 3 x^{13})^{5} d x

乘开

(x^{2} + 3 x^{13})^{5} : x^{10} + 15 x^{21} + 90 x^{32} + 270 x^{43} + 405 x^{54} + 243 x^{65}

= \int x^{10} + 15 x^{21} + 90 x^{32} + 270 x^{43} + 405 x^{54} + 243 x^{65} 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^{10} d x + \int 15 x^{21} d x + \int 90 x^{32} d x + \int 270 x^{43} d x + \int 405 x^{54} d x + \int 243 x^{65} d x

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

\int 15 x^{21} d x = \frac{15 x ^{22}}{22}

\int 90 x^{32} d x = \frac{30 x ^{33}}{11}

\int 270 x^{43} d x = \frac{135 x ^{44}}{22}

\int 405 x^{54} d x = \frac{81 x ^{55}}{11}

\int 243 x^{65} d x = \frac{81 x ^{66}}{22}

= \frac{x ^{11}}{11} + \frac{15 x ^{22}}{22} + \frac{30 x ^{33}}{11} + \frac{135 x ^{44}}{22} + \frac{81 x ^{55}}{11} + \frac{81 x ^{66}}{22}

解答补常数

= \frac{x ^{11}}{11} + \frac{15 x ^{22}}{22} + \frac{30 x ^{33}}{11} + \frac{135 x ^{44}}{22} + \frac{81 x ^{55}}{11} + \frac{81 x ^{66}}{22} + C

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