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 (8-3x)/((x+1)(x^2-4x+6))

解

\int \frac{8 - 3 x}{( x + 1 ) ( x ^{2} - 4 x + 6 )} d x

解

- \frac{1}{2} ln 9 x^{2} - 36 x + 54 + ln ∣ 3 x + 3 ∣ + C

解答ステップ

\int \frac{8 - 3 x}{( x + 1 ) ( x ^{2} - 4 x + 6 )} d x

u置換積分法を適用する

= \int - \frac{9 u}{( u ^{2} - 4 u + 22 ) ( - u + 11 )} d u

定数を除く:

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

= - 9 \cdot \int \frac{u}{( u ^{2} - 4 u + 22 ) ( - u + 11 )} d u

以下の部分分数を得る:

\frac{u}{( u ^{2} - 4 u + 22 ) ( - u + 11 )} : \frac{u - 2}{9 ( u ^{2} - 4 u + 22 )} - \frac{1}{9 ( u - 11 )}

= - 9 \cdot \int \frac{u - 2}{9 ( u ^{2} - 4 u + 22 )} - \frac{1}{9 ( u - 11 )} 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

= - 9 (\int \frac{u - 2}{9 ( u ^{2} - 4 u + 22 )} d u - \int \frac{1}{9 ( u - 11 )} d u)

\int \frac{u - 2}{9 ( u ^{2} - 4 u + 22 )} d u = \frac{1}{18} ln u^{2} - 4 u + 22

\int \frac{1}{9 ( u - 11 )} d u = \frac{1}{9} ln ∣ u - 11 ∣

= - 9 (\frac{1}{18} ln u^{2} - 4 u + 22 - \frac{1}{9} ln ∣ u - 11 ∣)

代用を戻す

u = 8 - 3 x

= - 9 (\frac{1}{18} ln (8 - 3 x)^{2} - 4 (8 - 3 x) + 22 - \frac{1}{9} ln ∣ 8 - 3 x - 11 ∣)

簡素化

- 9 (\frac{1}{18} ln (8 - 3 x)^{2} - 4 (8 - 3 x) + 22 - \frac{1}{9} ln ∣ 8 - 3 x - 11 ∣) : - \frac{1}{2} ln 9 x^{2} - 36 x + 54 + ln ∣ 3 x + 3 ∣

= - \frac{1}{2} ln 9 x^{2} - 36 x + 54 + ln ∣ 3 x + 3 ∣

定数を解答に追加する

= - \frac{1}{2} ln 9 x^{2} - 36 x + 54 + ln ∣ 3 x + 3 ∣ + C

グラフ

人気の例

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

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

integral of 2cos^3(x)sqrt(sin(x))

\int 2 cos^{3} (x) sin (x) d x

integral of (2x)/(sqrt(x^4+9))

\int \frac{2 x}{x ^{4} + 9} d x

integral of (sin(4e^{-x}))/(e^x)

\int \frac{sin ( 4 e ^{- x} )}{e ^{x}} d x

integral of 3/5 (v-2)e^{2-4v+v^2}

\int \frac{3}{5} (v - 2) e^{2 - 4 v + v^{2}} d v