integral from 4 to 6 of (3x-4)/(x^2-x-6)

解答

\int_{4}^{6} \frac{3 x - 4}{x ^{2} - x - 6} d x

2 ln (\frac{4}{3}) + ln (3)

十进制

1.67397 \dots

求解步骤

\int_{4}^{6} \frac{3 x - 4}{x ^{2} - x - 6} d x

将

\frac{3 x - 4}{x ^{2} - x - 6}

用部份分式展开:

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

= \int_{4}^{6} \frac{2}{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}^{6} \frac{2}{x + 2} d x + \int_{4}^{6} \frac{1}{x - 3} d x

\int_{4}^{6} \frac{2}{x + 2} d x = 2 ln (\frac{4}{3})

\int_{4}^{6} \frac{1}{x - 3} d x = ln (3)

= 2 ln (\frac{4}{3}) + ln (3)

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