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 5/(x(x+1)^2)

解

\int \frac{5}{x ( x + 1 ) ^{2}} d x

解

5 (ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1}) + C

解答ステップ

\int \frac{5}{x ( x + 1 ) ^{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{1}{x ( x + 1 ) ^{2}} d x

u置換積分法を適用する

= 5 \cdot \int \frac{1}{( u - 1 ) u ^{2}} d u

以下の部分分数を得る:

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

= 5 \cdot \int \frac{1}{u - 1} - \frac{1}{u} - \frac{1}{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 - 1} d u - \int \frac{1}{u} d u - \int \frac{1}{u ^{2}} d u)

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

\int \frac{1}{u} d u = ln ∣ u ∣

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

= 5 (ln ∣ u - 1 ∣ - ln ∣ u ∣ - (- \frac{1}{u}))

代用を戻す

u = x + 1

= 5 (ln ∣ x + 1 - 1 ∣ - ln ∣ x + 1 ∣ - (- \frac{1}{x + 1}))

簡素化

= 5 (ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1})

定数を解答に追加する

= 5 (ln ∣ x ∣ - ln ∣ x + 1 ∣ + \frac{1}{x + 1}) + C

グラフ

人気の例

integral of 50-3xe^{0.01x^2}

\int 50 - 3 x e^{0.01 x^{2}} d x

integral of (21t^2-10t)/(7t^3-5t^2+13)

\int \frac{21 t ^{2} - 10 t}{7 t ^{3} - 5 t ^{2} + 13} d t

integral of (x^2+2)/(sqrt(x^3+6x))

\int \frac{x ^{2} + 2}{x ^{3} + 6 x} d x

integral of sin^5(v)cos^2(v)

\int sin^{5} (v) cos^{2} (v) d v

integral of (csc(x))/(1+sec(x))

\int \frac{csc ( x )}{1 + sec ( x )} d x