integral of x^4(5x^5+9)^7

解答

\int x^{4} (5 x^{5} + 9)^{7} d x

\frac{15625 x ^{40}}{8} + 28125 x^{35} + \frac{354375 x ^{30}}{2} + 637875 x^{25} + \frac{5740875 x ^{20}}{4} + 2066715 x^{15} + \frac{3720087 x ^{10}}{2} + \frac{4782969 x ^{5}}{5} + C

求解步骤

\int x^{4} (5 x^{5} + 9)^{7} d x

乘开

x^{4} (5 x^{5} + 9)^{7} : 78125 x^{39} + 984375 x^{34} + 5315625 x^{29} + 15946875 x^{24} + 28704375 x^{19} + 31000725 x^{14} + 18600435 x^{9} + 4782969 x^{4}

= \int 78125 x^{39} + 984375 x^{34} + 5315625 x^{29} + 15946875 x^{24} + 28704375 x^{19} + 31000725 x^{14} + 18600435 x^{9} + 4782969 x^{4} 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^{39} d x + \int 984375 x^{34} d x + \int 5315625 x^{29} d x + \int 15946875 x^{24} d x + \int 28704375 x^{19} d x + \int 31000725 x^{14} d x + \int 18600435 x^{9} d x + \int 4782969 x^{4} d x

\int 78125 x^{39} d x = \frac{15625 x ^{40}}{8}

\int 984375 x^{34} d x = 28125 x^{35}

\int 5315625 x^{29} d x = \frac{354375 x ^{30}}{2}

\int 15946875 x^{24} d x = 637875 x^{25}

\int 28704375 x^{19} d x = \frac{5740875 x ^{20}}{4}

\int 31000725 x^{14} d x = 2066715 x^{15}

\int 18600435 x^{9} d x = \frac{3720087 x ^{10}}{2}

\int 4782969 x^{4} d x = \frac{4782969 x ^{5}}{5}

= \frac{15625 x ^{40}}{8} + 28125 x^{35} + \frac{354375 x ^{30}}{2} + 637875 x^{25} + \frac{5740875 x ^{20}}{4} + 2066715 x^{15} + \frac{3720087 x ^{10}}{2} + \frac{4782969 x ^{5}}{5}

解答补常数

= \frac{15625 x ^{40}}{8} + 28125 x^{35} + \frac{354375 x ^{30}}{2} + 637875 x^{25} + \frac{5740875 x ^{20}}{4} + 2066715 x^{15} + \frac{3720087 x ^{10}}{2} + \frac{4782969 x ^{5}}{5} + C

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