integral of (x^4-5x)^2(4x^3-5)

解答

\int (x^{4} - 5 x)^{2} (4 x^{3} - 5) d x

\frac{x ^{12}}{3} - 5 x^{9} + 25 x^{6} - \frac{125 x ^{3}}{3} + C

求解步骤

\int (x^{4} - 5 x)^{2} (4 x^{3} - 5) d x

乘开

(x^{4} - 5 x)^{2} (4 x^{3} - 5) : 4 x^{11} - 45 x^{8} + 150 x^{5} - 125 x^{2}

= \int 4 x^{11} - 45 x^{8} + 150 x^{5} - 125 x^{2} 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 4 x^{11} d x - \int 45 x^{8} d x + \int 150 x^{5} d x - \int 125 x^{2} d x

\int 4 x^{11} d x = \frac{x ^{12}}{3}

\int 45 x^{8} d x = 5 x^{9}

\int 150 x^{5} d x = 25 x^{6}

\int 125 x^{2} d x = \frac{125 x ^{3}}{3}

= \frac{x ^{12}}{3} - 5 x^{9} + 25 x^{6} - \frac{125 x ^{3}}{3}

解答补常数

= \frac{x ^{12}}{3} - 5 x^{9} + 25 x^{6} - \frac{125 x ^{3}}{3} + C

\int \frac{2 x}{2 - x ^{2}} d x

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