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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

以下の部分分数を得る:

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

= 3 \cdot \int - \frac{1}{x} - \frac{1}{x ^{2}} + \frac{1}{x - 1} 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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

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

簡素化

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

定数を解答に追加する

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

グラフ

人気の例

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