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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

u置換積分法を適用する

= \int \frac{u + 2}{u ( u + 1 ) ^{2}} d u

拡張

\frac{u + 2}{u ( u + 1 ) ^{2}} : \frac{u}{u ( u + 1 ) ^{2}} + \frac{2}{u ( u + 1 ) ^{2}}

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{u}{u ( u + 1 ) ^{2}} d u + \int \frac{2}{u ( u + 1 ) ^{2}} d u

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

\int \frac{2}{u ( u + 1 ) ^{2}} d u = 2 (ln ∣ u ∣ - ln ∣ u + 1 ∣ + \frac{1}{u + 1})

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

代用を戻す

u = x - 2

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

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

integral of ((2x+2))/(x^2+10x+24)

\int \frac{( 2 x + 2 )}{x ^{2} + 10 x + 24} d x

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

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

integral of ln(x)\sqrt[3]{x}

\int ln (x) 3 x d x

integral of 5(z-4)\sqrt[3]{z^2-8z}

\int 5 (z - 4) 3 z^{2} - 8 z d z

integral of (8s^3)/(sqrt(5-s^4))

\int \frac{8 s ^{3}}{5 - s ^{4}} d s