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

解答

\int_{- 3}^{- 2} \frac{5 x + 2}{x ^{2} + x} d x

- ln (2) - 2 ln (3)

十进制

- 2.89037 \dots

求解步骤

\int_{- 3}^{- 2} \frac{5 x + 2}{x ^{2} + x} d x

将

\frac{5 x + 2}{x ^{2} + x}

用部份分式展开:

\frac{2}{x} + \frac{3}{x + 1}

= \int_{- 3}^{- 2} \frac{2}{x} + \frac{3}{x + 1} 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_{- 3}^{- 2} \frac{2}{x} d x + \int_{- 3}^{- 2} \frac{3}{x + 1} d x

\int_{- 3}^{- 2} \frac{2}{x} d x = 2 (ln (2) - ln (3))

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

= 2 (ln (2) - ln (3)) - 3 ln (2)

化简

2 (ln (2) - ln (3)) - 3 ln (2) : - ln (2) - 2 ln (3)

= - ln (2) - 2 ln (3)

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