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人気のある微分積分 >

integral of (400)/(x^3-20x^2)

解

\int \frac{400}{x ^{3} - 20 x ^{2}} d x

解

- ln ∣ x ∣ + \frac{20}{x} + ln ∣ x - 20 ∣ + C

解答ステップ

\int \frac{400}{x ^{3} - 20 x ^{2}} d x

定数を除く:

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

= 400 \cdot \int \frac{1}{x ^{3} - 20 x ^{2}} d x

以下の部分分数を得る:

\frac{1}{x ^{3} - 20 x ^{2}} : - \frac{1}{400 x} - \frac{1}{20 x ^{2}} + \frac{1}{400 ( x - 20 )}

= 400 \cdot \int - \frac{1}{400 x} - \frac{1}{20 x ^{2}} + \frac{1}{400 ( x - 20 )} 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

= 400 (- \int \frac{1}{400 x} d x - \int \frac{1}{20 x ^{2}} d x + \int \frac{1}{400 ( x - 20 )} d x)

\int \frac{1}{400 x} d x = \frac{1}{400} ln ∣ x ∣

\int \frac{1}{20 x ^{2}} d x = - \frac{1}{20 x}

\int \frac{1}{400 ( x - 20 )} d x = \frac{1}{400} ln ∣ x - 20 ∣

= 400 (- \frac{1}{400} ln ∣ x ∣ - (- \frac{1}{20 x}) + \frac{1}{400} ln ∣ x - 20 ∣)

簡素化

400 (- \frac{1}{400} ln ∣ x ∣ - (- \frac{1}{20 x}) + \frac{1}{400} ln ∣ x - 20 ∣) : - ln ∣ x ∣ + \frac{20}{x} + ln ∣ x - 20 ∣

= - ln ∣ x ∣ + \frac{20}{x} + ln ∣ x - 20 ∣

定数を解答に追加する

= - ln ∣ x ∣ + \frac{20}{x} + ln ∣ x - 20 ∣ + C

グラフ

人気の例

integral from 4 to 16 of 1/(1-sqrt(x))

\int_{4}^{16} \frac{1}{1 - x} d x

tangent f(x)=4x^2+4x,\at x=-2

t an g e n t f (x) = 4 x^{2} + 4 x, at x = - 2

integral of e^{-1/4 x^2}

\int e^{- \frac{1}{4} x^{2}} d x

derivative of 1-(1-x/(40)^3)

\frac{d}{d x} (1 - (1 - \frac{x}{40})^{3})

(dy)/(dx)= x/(x^2+1)

\frac{d y}{d x} = \frac{x}{x ^{2} + 1}