integral of (5x-1)/(x(x-3)^2(x+2)^2)

解答

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

- \frac{1}{36} ln ∣ x ∣ - \frac{79}{1125} ln ∣ x - 3 ∣ - \frac{14}{75 ( x - 3 )} + \frac{49}{500} ln ∣ x + 2 ∣ - \frac{11}{50 ( x + 2 )} + C

求解步骤

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

将

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

用部份分式展开:

- \frac{1}{36 x} - \frac{79}{1125 ( x - 3 )} + \frac{14}{75 ( x - 3 ) ^{2}} + \frac{49}{500 ( x + 2 )} + \frac{11}{50 ( x + 2 ) ^{2}}

= \int - \frac{1}{36 x} - \frac{79}{1125 ( x - 3 )} + \frac{14}{75 ( x - 3 ) ^{2}} + \frac{49}{500 ( x + 2 )} + \frac{11}{50 ( x + 2 ) ^{2}} 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}{36 x} d x - \int \frac{79}{1125 ( x - 3 )} d x + \int \frac{14}{75 ( x - 3 ) ^{2}} d x + \int \frac{49}{500 ( x + 2 )} d x + \int \frac{11}{50 ( x + 2 ) ^{2}} d x

\int \frac{1}{36 x} d x = \frac{1}{36} ln ∣ x ∣

\int \frac{79}{1125 ( x - 3 )} d x = \frac{79}{1125} ln ∣ x - 3 ∣

\int \frac{14}{75 ( x - 3 ) ^{2}} d x = - \frac{14}{75 ( x - 3 )}

\int \frac{49}{500 ( x + 2 )} d x = \frac{49}{500} ln ∣ x + 2 ∣

\int \frac{11}{50 ( x + 2 ) ^{2}} d x = - \frac{11}{50 ( x + 2 )}

= - \frac{1}{36} ln ∣ x ∣ - \frac{79}{1125} ln ∣ x - 3 ∣ - \frac{14}{75 ( x - 3 )} + \frac{49}{500} ln ∣ x + 2 ∣ - \frac{11}{50 ( x + 2 )}

解答补常数

= - \frac{1}{36} ln ∣ x ∣ - \frac{79}{1125} ln ∣ x - 3 ∣ - \frac{14}{75 ( x - 3 )} + \frac{49}{500} ln ∣ x + 2 ∣ - \frac{11}{50 ( x + 2 )} + C

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