integral of (x-1)/((x-2)(x+3))

解答

\int \frac{x - 1}{( x - 2 ) ( x + 3 )} d x

\frac{1}{5} ln ∣ x - 2 ∣ + \frac{4}{5} ln ∣ x + 3 ∣ + C

求解步骤

\int \frac{x - 1}{( x - 2 ) ( x + 3 )} d x

将

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

用部份分式展开:

\frac{1}{5 ( x - 2 )} + \frac{4}{5 ( x + 3 )}

= \int \frac{1}{5 ( x - 2 )} + \frac{4}{5 ( 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{1}{5 ( x - 2 )} d x + \int \frac{4}{5 ( x + 3 )} d x

\int \frac{1}{5 ( x - 2 )} d x = \frac{1}{5} ln ∣ x - 2 ∣

\int \frac{4}{5 ( x + 3 )} d x = \frac{4}{5} ln ∣ x + 3 ∣

= \frac{1}{5} ln ∣ x - 2 ∣ + \frac{4}{5} ln ∣ x + 3 ∣

解答补常数

= \frac{1}{5} ln ∣ x - 2 ∣ + \frac{4}{5} ln ∣ x + 3 ∣ + C

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