integral from-2 to 3 of (5x^2-3x-2)^2

解答

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

\frac{4855}{6}

十进制

809.16666 \dots

求解步骤

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

乘开

(5 x^{2} - 3 x - 2)^{2} : 25 x^{4} - 30 x^{3} - 11 x^{2} + 12 x + 4

= \int_{- 2}^{3} 25 x^{4} - 30 x^{3} - 11 x^{2} + 12 x + 4 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_{- 2}^{3} 25 x^{4} d x - \int_{- 2}^{3} 30 x^{3} d x - \int_{- 2}^{3} 11 x^{2} d x + \int_{- 2}^{3} 12 x d x + \int_{- 2}^{3} 4 d x

\int_{- 2}^{3} 25 x^{4} d x = 1375

\int_{- 2}^{3} 30 x^{3} d x = \frac{975}{2}

\int_{- 2}^{3} 11 x^{2} d x = \frac{385}{3}

\int_{- 2}^{3} 12 x d x = 30

\int_{- 2}^{3} 4 d x = 20

= 1375 - \frac{975}{2} - \frac{385}{3} + 30 + 20

化简

1375 - \frac{975}{2} - \frac{385}{3} + 30 + 20 : \frac{4855}{6}

= \frac{4855}{6}

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