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人気のある微分積分 >

integral of (121)/(x^3-11x^2)

解

\int \frac{121}{x ^{3} - 11 x ^{2}} d x

解

- ln ∣ x ∣ + \frac{11}{x} + ln ∣ x - 11 ∣ + C

解答ステップ

\int \frac{121}{x ^{3} - 11 x ^{2}} d x

定数を除く:

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

= 121 \cdot \int \frac{1}{x ^{3} - 11 x ^{2}} d x

以下の部分分数を得る:

\frac{1}{x ^{3} - 11 x ^{2}} : - \frac{1}{121 x} - \frac{1}{11 x ^{2}} + \frac{1}{121 ( x - 11 )}

= 121 \cdot \int - \frac{1}{121 x} - \frac{1}{11 x ^{2}} + \frac{1}{121 ( x - 11 )} 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

= 121 (- \int \frac{1}{121 x} d x - \int \frac{1}{11 x ^{2}} d x + \int \frac{1}{121 ( x - 11 )} d x)

\int \frac{1}{121 x} d x = \frac{1}{121} ln ∣ x ∣

\int \frac{1}{11 x ^{2}} d x = - \frac{1}{11 x}

\int \frac{1}{121 ( x - 11 )} d x = \frac{1}{121} ln ∣ x - 11 ∣

= 121 (- \frac{1}{121} ln ∣ x ∣ - (- \frac{1}{11 x}) + \frac{1}{121} ln ∣ x - 11 ∣)

簡素化

121 (- \frac{1}{121} ln ∣ x ∣ - (- \frac{1}{11 x}) + \frac{1}{121} ln ∣ x - 11 ∣) : - ln ∣ x ∣ + \frac{11}{x} + ln ∣ x - 11 ∣

= - ln ∣ x ∣ + \frac{11}{x} + ln ∣ x - 11 ∣

定数を解答に追加する

= - ln ∣ x ∣ + \frac{11}{x} + ln ∣ x - 11 ∣ + C

グラフ

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