integral of x^3(x^4+5)^3

解答

\int x^{3} (x^{4} + 5)^{3} d x

\frac{x ^{16}}{16} + \frac{5 x ^{12}}{4} + \frac{75 x ^{8}}{8} + \frac{125 x ^{4}}{4} + C

求解步骤

\int x^{3} (x^{4} + 5)^{3} d x

乘开

x^{3} (x^{4} + 5)^{3} : x^{15} + 15 x^{11} + 75 x^{7} + 125 x^{3}

= \int x^{15} + 15 x^{11} + 75 x^{7} + 125 x^{3} 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^{15} d x + \int 15 x^{11} d x + \int 75 x^{7} d x + \int 125 x^{3} d x

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

\int 15 x^{11} d x = \frac{5 x ^{12}}{4}

\int 75 x^{7} d x = \frac{75 x ^{8}}{8}

\int 125 x^{3} d x = \frac{125 x ^{4}}{4}

= \frac{x ^{16}}{16} + \frac{5 x ^{12}}{4} + \frac{75 x ^{8}}{8} + \frac{125 x ^{4}}{4}

解答补常数

= \frac{x ^{16}}{16} + \frac{5 x ^{12}}{4} + \frac{75 x ^{8}}{8} + \frac{125 x ^{4}}{4} + C

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