integral of 2x(6x^2-12)^2

解答

\int 2 x (6 x^{2} - 12)^{2} d x

2 (6 x^{6} - 36 x^{4} + 72 x^{2}) + C

求解步骤

\int 2 x (6 x^{2} - 12)^{2} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x (6 x^{2} - 12)^{2} d x

乘开

x (6 x^{2} - 12)^{2} : 36 x^{5} - 144 x^{3} + 144 x

= 2 \cdot \int 36 x^{5} - 144 x^{3} + 144 x 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

= 2 (\int 36 x^{5} d x - \int 144 x^{3} d x + \int 144 x d x)

\int 36 x^{5} d x = 6 x^{6}

\int 144 x^{3} d x = 36 x^{4}

\int 144 x d x = 72 x^{2}

= 2 (6 x^{6} - 36 x^{4} + 72 x^{2})

解答补常数

= 2 (6 x^{6} - 36 x^{4} + 72 x^{2}) + C

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