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

解

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

\frac{1}{36} ln ∣ x ∣ - \frac{11}{1125} ln ∣ x - 3 ∣ - \frac{1}{75 ( x - 3 )} - \frac{9}{500} ln ∣ x + 2 ∣ + \frac{1}{50 ( x + 2 )} + C

解答ステップ

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

以下の部分分数を得る:

\frac{1}{x ( x - 3 ) ^{2} ( x + 2 ) ^{2}} : \frac{1}{36 x} - \frac{11}{1125 ( x - 3 )} + \frac{1}{75 ( x - 3 ) ^{2}} - \frac{9}{500 ( x + 2 )} - \frac{1}{50 ( x + 2 ) ^{2}}

= \int \frac{1}{36 x} - \frac{11}{1125 ( x - 3 )} + \frac{1}{75 ( x - 3 ) ^{2}} - \frac{9}{500 ( x + 2 )} - \frac{1}{50 ( x + 2 ) ^{2}} d x

総和規則を適用する:

\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{11}{1125 ( x - 3 )} d x + \int \frac{1}{75 ( x - 3 ) ^{2}} d x - \int \frac{9}{500 ( x + 2 )} d x - \int \frac{1}{50 ( x + 2 ) ^{2}} d x

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

\int \frac{11}{1125 ( x - 3 )} d x = \frac{11}{1125} ln ∣ x - 3 ∣

\int \frac{1}{75 ( x - 3 ) ^{2}} d x = - \frac{1}{75 ( x - 3 )}

\int \frac{9}{500 ( x + 2 )} d x = \frac{9}{500} ln ∣ x + 2 ∣

\int \frac{1}{50 ( x + 2 ) ^{2}} d x = - \frac{1}{50 ( x + 2 )}

= \frac{1}{36} ln ∣ x ∣ - \frac{11}{1125} ln ∣ x - 3 ∣ - \frac{1}{75 ( x - 3 )} - \frac{9}{500} ln ∣ x + 2 ∣ - (- \frac{1}{50 ( x + 2 )})

簡素化

= \frac{1}{36} ln ∣ x ∣ - \frac{11}{1125} ln ∣ x - 3 ∣ - \frac{1}{75 ( x - 3 )} - \frac{9}{500} ln ∣ x + 2 ∣ + \frac{1}{50 ( x + 2 )}

定数を解答に追加する

= \frac{1}{36} ln ∣ x ∣ - \frac{11}{1125} ln ∣ x - 3 ∣ - \frac{1}{75 ( x - 3 )} - \frac{9}{500} ln ∣ x + 2 ∣ + \frac{1}{50 ( x + 2 )} + C