integral from 4 to 5 of 1/(x^2-5x+6)

解答

\int_{4}^{5} \frac{1}{x ^{2} - 5 x + 6} d x

- ln (3) + 2 ln (2)

十进制

0.28768 \dots

求解步骤

\int_{4}^{5} \frac{1}{x ^{2} - 5 x + 6} d x

将

\frac{1}{x ^{2} - 5 x + 6}

用部份分式展开:

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

= \int_{4}^{5} - \frac{1}{x - 2} + \frac{1}{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_{4}^{5} \frac{1}{x - 2} d x + \int_{4}^{5} \frac{1}{x - 3} d x

\int_{4}^{5} \frac{1}{x - 2} d x = ln (3) - ln (2)

\int_{4}^{5} \frac{1}{x - 3} d x = ln (2)

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

化简

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

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