integral from-2 to 3 of (x-1)(x+3)

解答

\int_{- 2}^{3} (x - 1) (x + 3) d x

\frac{5}{3}

十进制

1.66666 \dots

求解步骤

\int_{- 2}^{3} (x - 1) (x + 3) d x

乘开

(x - 1) (x + 3) : x^{2} + 2 x - 3

= \int_{- 2}^{3} x^{2} + 2 x - 3 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} x^{2} d x + \int_{- 2}^{3} 2 x d x - \int_{- 2}^{3} 3 d x

\int_{- 2}^{3} x^{2} d x = \frac{35}{3}

\int_{- 2}^{3} 2 x d x = 5

\int_{- 2}^{3} 3 d x = 15

= \frac{35}{3} + 5 - 15

化简

\frac{35}{3} + 5 - 15 : \frac{5}{3}

= \frac{5}{3}

\int_{0}^{1} \frac{5}{( 2 x + 1 ) ^{2}} d x

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