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 4/(x(x-3)(x+1)^2)

解

\int \frac{4}{x ( x - 3 ) ( x + 1 ) ^{2}} d x

解

- \frac{4}{3} ln ∣ x ∣ + \frac{1}{12} ln ∣ x - 3 ∣ + \frac{5}{4} ln ∣ x + 1 ∣ - \frac{1}{x + 1} + C

解答ステップ

\int \frac{4}{x ( x - 3 ) ( x + 1 ) ^{2}} d x

定数を除く:

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

= 4 \cdot \int \frac{1}{x ( x - 3 ) ( x + 1 ) ^{2}} d x

以下の部分分数を得る:

\frac{1}{x ( x - 3 ) ( x + 1 ) ^{2}} : - \frac{1}{3 x} + \frac{1}{48 ( x - 3 )} + \frac{5}{16 ( x + 1 )} + \frac{1}{4 ( x + 1 ) ^{2}}

= 4 \cdot \int - \frac{1}{3 x} + \frac{1}{48 ( x - 3 )} + \frac{5}{16 ( x + 1 )} + \frac{1}{4 ( x + 1 ) ^{2}} 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

= 4 (- \int \frac{1}{3 x} d x + \int \frac{1}{48 ( x - 3 )} d x + \int \frac{5}{16 ( x + 1 )} d x + \int \frac{1}{4 ( x + 1 ) ^{2}} d x)

\int \frac{1}{3 x} d x = \frac{1}{3} ln ∣ x ∣

\int \frac{1}{48 ( x - 3 )} d x = \frac{1}{48} ln ∣ x - 3 ∣

\int \frac{5}{16 ( x + 1 )} d x = \frac{5}{16} ln ∣ x + 1 ∣

\int \frac{1}{4 ( x + 1 ) ^{2}} d x = - \frac{1}{4 ( x + 1 )}

= 4 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{48} ln ∣ x - 3 ∣ + \frac{5}{16} ln ∣ x + 1 ∣ - \frac{1}{4 ( x + 1 )})

簡素化

4 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{48} ln ∣ x - 3 ∣ + \frac{5}{16} ln ∣ x + 1 ∣ - \frac{1}{4 ( x + 1 )}) : - \frac{4}{3} ln ∣ x ∣ + \frac{1}{12} ln ∣ x - 3 ∣ + \frac{5}{4} ln ∣ x + 1 ∣ - \frac{1}{x + 1}

= - \frac{4}{3} ln ∣ x ∣ + \frac{1}{12} ln ∣ x - 3 ∣ + \frac{5}{4} ln ∣ x + 1 ∣ - \frac{1}{x + 1}

定数を解答に追加する

= - \frac{4}{3} ln ∣ x ∣ + \frac{1}{12} ln ∣ x - 3 ∣ + \frac{5}{4} ln ∣ x + 1 ∣ - \frac{1}{x + 1} + C

グラフ

人気の例

integral of (x-3)/(x^2+3x-4)

\int \frac{x - 3}{x ^{2} + 3 x - 4} d x

integral of (tan^3(θ))/(sec^4(θ))

\int \frac{tan ^{3} ( θ )}{sec ^{4} ( θ )} d θ

integral of x^2*cos(n*x)

\int x^{2} \cdot cos (n \cdot x) d x

integral of 2x^5+7+1/(x^3)

\int 2 x^{5} + 7 + \frac{1}{x ^{3}} d x

integral of (3e^{3x})/(1+e^{3x)}

\int \frac{3 e ^{3 x}}{1 + e ^{3 x}} d x