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 (30)/(x^2+x-6)

解

\int \frac{30}{x ^{2} + x - 6} d x

解

- 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x) + C

解答ステップ

\int \frac{30}{x ^{2} + x - 6} d x

定数を除く:

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

= 30 \cdot \int \frac{1}{x ^{2} + x - 6} d x

平方完成する

x^{2} + x - 6 : (x + \frac{1}{2})^{2} - \frac{25}{4}

= 30 \cdot \int \frac{1}{( x + \frac{1}{2} ) ^{2} - \frac{25}{4}} d x

u置換積分法を適用する

= 30 \cdot \int \frac{4}{4 u ^{2} - 25} d u

定数を除く:

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

= 30 \cdot 4 \cdot \int \frac{1}{4 u ^{2} - 25} d u

置換積分法を適用する

= 30 \cdot 4 \cdot \int \frac{1}{10 ( v ^{2} - 1 )} d v

定数を除く:

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

= 30 \cdot 4 \cdot \frac{1}{10} \cdot \int \frac{1}{v ^{2} - 1} d v

因数

v^{2} - 1 : - (- v^{2} + 1)

= 30 \cdot 4 \cdot \frac{1}{10} \cdot \int \frac{1}{- ( - v ^{2} + 1 )} d v

定数を除く:

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

= 30 \cdot 4 \cdot \frac{1}{10} (- \int \frac{1}{- v ^{2} + 1} d v)

共通積分を使用する:

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= 30 \cdot 4 \cdot \frac{1}{10} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

代用を戻す

= 30 \cdot 4 \cdot \frac{1}{10} (- (\frac{ln \frac{2}{5} ( x + \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{5} ( x + \frac{1}{2} ) - 1}{2}))

簡素化

30 \cdot 4 \cdot \frac{1}{10} (- (\frac{ln \frac{2}{5} ( x + \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{5} ( x + \frac{1}{2} ) - 1}{2})) : - 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x)

= - 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x)

定数を解答に追加する

= - 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x) + C

グラフ

人気の例

integral of sec^2(4x)e^{tan(4x)}

\int sec^{2} (4 x) e^{t a n (4 x)} d x

integral of (x^2-3x+2)(cos(x))

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

integral of sqrt(2x-4)

\int 2 x - 4 d x

integral of (1-2x)(3+x)

\int (1 - 2 x) (3 + x) d x

積分記号 sec^2(t)

in t e g r a l sec^{2} (t)