integral of (x-48)/(x^2-x-6)

解答

\int \frac{x - 48}{x ^{2} - x - 6} d x

10 ln ∣ x + 2 ∣ - 9 ln ∣ x - 3 ∣ + C

求解步骤

\int \frac{x - 48}{x ^{2} - x - 6} d x

将

\frac{x - 48}{x ^{2} - x - 6}

用部份分式展开:

\frac{10}{x + 2} - \frac{9}{x - 3}

= \int \frac{10}{x + 2} - \frac{9}{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 \frac{10}{x + 2} d x - \int \frac{9}{x - 3} d x

\int \frac{10}{x + 2} d x = 10 ln ∣ x + 2 ∣

\int \frac{9}{x - 3} d x = 9 ln ∣ x - 3 ∣

= 10 ln ∣ x + 2 ∣ - 9 ln ∣ x - 3 ∣

解答补常数

= 10 ln ∣ x + 2 ∣ - 9 ln ∣ x - 3 ∣ + C

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