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人気のある微分積分 >

integral of (5x^2)/(x(x-4)^2)

解

\int \frac{5 x ^{2}}{x ( x - 4 ) ^{2}} d x

解

5 (ln ∣ x - 4 ∣ - \frac{4}{x - 4}) + C

解答ステップ

\int \frac{5 x ^{2}}{x ( x - 4 ) ^{2}} d x

定数を除く:

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

= 5 \cdot \int \frac{x ^{2}}{x ( x - 4 ) ^{2}} d x

共通因数を約分する:

x

= 5 \cdot \int \frac{x}{( x - 4 ) ^{2}} d x

u置換積分法を適用する

= 5 \cdot \int \frac{u + 4}{u ^{2}} d u

拡張

\frac{u + 4}{u ^{2}} : \frac{1}{u} + \frac{4}{u ^{2}}

= 5 \cdot \int \frac{1}{u} + \frac{4}{u ^{2}} d u

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

= 5 (\int \frac{1}{u} d u + \int \frac{4}{u ^{2}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{4}{u ^{2}} d u = - \frac{4}{u}

= 5 (ln ∣ u ∣ - \frac{4}{u})

代用を戻す

u = x - 4

= 5 (ln ∣ x - 4 ∣ - \frac{4}{x - 4})

定数を解答に追加する

= 5 (ln ∣ x - 4 ∣ - \frac{4}{x - 4}) + C

グラフ

人気の例

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\int_{0}^{2} 2 - x^{2} d x

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\int_{- 2}^{1} \frac{1}{( 11 + 5 x ) ^{3}} d x

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\int_{- \frac{1}{8}}^{\frac{1}{8}} sin (2 π x) d x

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