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

解答

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

5 x^{5} - 155 x^{4} + 90 x^{3} - 545 x^{2} + 81 x + C

求解步骤

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

乘开

(5 x - 3)^{4} : 25 x^{4} - 12 \cdot 5^{\frac{3}{2}} x^{3} + 270 x^{2} - 1085 x + 81

= \int 25 x^{4} - 12 \cdot 5^{\frac{3}{2}} x^{3} + 270 x^{2} - 1085 x + 81 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 25 x^{4} d x - \int 12 \cdot 5^{\frac{3}{2}} x^{3} d x + \int 270 x^{2} d x - \int 1085 x d x + \int 81 d x

\int 25 x^{4} d x = 5 x^{5}

\int 12 \cdot 5^{\frac{3}{2}} x^{3} d x = 155 x^{4}

\int 270 x^{2} d x = 90 x^{3}

\int 1085 x d x = 545 x^{2}

\int 81 d x = 81 x

= 5 x^{5} - 155 x^{4} + 90 x^{3} - 545 x^{2} + 81 x

解答补常数

= 5 x^{5} - 155 x^{4} + 90 x^{3} - 545 x^{2} + 81 x + C

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