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

解

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

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

解答ステップ

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

以下の部分分数を得る:

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

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

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

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

\int \frac{2}{x - 3} d x = 2 ln ∣ x - 3 ∣

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

簡素化

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

定数を解答に追加する

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