integral from-3 to 3 of (t^3-9t)(-20t)

解答

\int_{- 3}^{3} (t^{3} - 9 t) (- 20 t) d t

1296

求解步骤

\int_{- 3}^{3} (t^{3} - 9 t) (- 20 t) d t

化简

= \int_{- 3}^{3} - 20 t (t^{3} - 9 t) d t

提出常数:

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

= - 20 \cdot \int_{- 3}^{3} t (t^{3} - 9 t) d t

乘开

t (t^{3} - 9 t) : t^{4} - 9 t^{2}

= - 20 \cdot \int_{- 3}^{3} t^{4} - 9 t^{2} d t

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

= - 20 (\int_{- 3}^{3} t^{4} d t - \int_{- 3}^{3} 9 t^{2} d t)

\int_{- 3}^{3} t^{4} d t = \frac{486}{5}

\int_{- 3}^{3} 9 t^{2} d t = 162

= - 20 (\frac{486}{5} - 162)

化简

- 20 (\frac{486}{5} - 162) : 1296

= 1296

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