ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of (37)/(x^3-125)

解

\int \frac{37}{x ^{3} - 125} d x

解

37 (\frac{1}{75} ln ∣ x - 5 ∣ - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3})) + C

解答ステップ

\int \frac{37}{x ^{3} - 125} d x

定数を除く:

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

= 37 \cdot \int \frac{1}{x ^{3} - 125} d x

以下の部分分数を得る:

\frac{1}{x ^{3} - 125} : \frac{1}{75 ( x - 5 )} + \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )}

= 37 \cdot \int \frac{1}{75 ( x - 5 )} + \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} 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

= 37 (\int \frac{1}{75 ( x - 5 )} d x + \int \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} d x)

\int \frac{1}{75 ( x - 5 )} d x = \frac{1}{75} ln ∣ x - 5 ∣

\int \frac{- x - 10}{75 ( x ^{2} + 5 x + 25 )} d x = \frac{1}{150 3} (- 3 ln 4 x^{2} + 20 x + 100 - 6 arctan (\frac{2}{5 3} x + \frac{1}{3}))

= 37 (\frac{1}{75} ln ∣ x - 5 ∣ + \frac{1}{150 3} (- 3 ln 4 x^{2} + 20 x + 100 - 6 arctan (\frac{2}{5 3} x + \frac{1}{3})))

拡張

\frac{1}{150 3} (- 3 ln 4 x^{2} + 20 x + 100 - 6 arctan (\frac{2}{5 3} x + \frac{1}{3})) : - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3})

= 37 (\frac{1}{75} ln ∣ x - 5 ∣ - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3}))

定数を解答に追加する

= 37 (\frac{1}{75} ln ∣ x - 5 ∣ - \frac{1}{150} ln 4 x^{2} + 20 x + 100 - \frac{1}{25 3} arctan (\frac{2}{5 3} x + \frac{1}{3})) + C

グラフ

人気の例

derivative 3x^2sin(x)tan(x)

d er i v a t i v e 3 x^{2} sin (x) tan (x)

(dy)/(dx)-y/x =2xe^x,y(1)=2e-8

\frac{d y}{d x} - \frac{y}{x} = 2 x e^{x}, y (1) = 2 e - 8

derivative of \sqrt[13]{9+7\sqrt[5]{2x}}

\frac{d}{d x} (13 9 + 7 5 2 x)

integral of (sec^2(θ)+sec(θ)tan(θ))

\int (sec^{2} (θ) + sec (θ) tan (θ)) d θ

d/(dy)(sqrt(xy))

\frac{d}{d y} (x y)