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 ((3x-5))/(sqrt(2x^2-6x+3))

解

\int \frac{( 3 x - 5 )}{2 x ^{2} - 6 x + 3} d x

解

2 (\frac{3}{4} 4 x^{2} - 12 x + 6 - \frac{1}{4} ln \frac{4 x ^{2} - 12 x + 6}{3} + \frac{2}{3} x - 3) + C

解答ステップ

\int \frac{3 x - 5}{2 x ^{2} - 6 x + 3} d x

平方完成する

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

= \int \frac{3 x - 5}{2 ( x - \frac{3}{2} ) ^{2} - \frac{3}{2}} d x

u置換積分法を適用する

= \int \frac{2 ( 3 ( u + \frac{3}{2} ) - 5 )}{4 u ^{2} - 3} d u

定数を除く:

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

= 2 \cdot \int \frac{3 ( u + \frac{3}{2} ) - 5}{4 u ^{2} - 3} d u

拡張

\frac{3 ( u + \frac{3}{2} ) - 5}{4 u ^{2} - 3} : \frac{3 u}{4 u ^{2} - 3} - \frac{1}{2 4 u ^{2} - 3}

= 2 \cdot \int \frac{3 u}{4 u ^{2} - 3} - \frac{1}{2 4 u ^{2} - 3} 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

= 2 (\int \frac{3 u}{4 u ^{2} - 3} d u - \int \frac{1}{2 4 u ^{2} - 3} d u)

\int \frac{3 u}{4 u ^{2} - 3} d u = \frac{3}{4} 4 u^{2} - 3

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

= 2 (\frac{3}{4} 4 u^{2} - 3 - \frac{1}{4} ln \frac{1}{3} (4 u^{2} - 3) + \frac{2}{3} u)

代用を戻す

u = x - \frac{3}{2}

= 2 \frac{3}{4} 4 (x - \frac{3}{2})^{2} - 3 - \frac{1}{4} ln \frac{1}{3} (4 (x - \frac{3}{2})^{2} - 3) + \frac{2}{3} (x - \frac{3}{2})

簡素化

2 \frac{3}{4} 4 (x - \frac{3}{2})^{2} - 3 - \frac{1}{4} ln \frac{1}{3} (4 (x - \frac{3}{2})^{2} - 3) + \frac{2}{3} (x - \frac{3}{2}) : 2 (\frac{3}{4} 4 x^{2} - 12 x + 6 - \frac{1}{4} ln \frac{4 x ^{2} - 12 x + 6}{3} + \frac{2}{3} x - 3)

= 2 (\frac{3}{4} 4 x^{2} - 12 x + 6 - \frac{1}{4} ln \frac{4 x ^{2} - 12 x + 6}{3} + \frac{2}{3} x - 3)

定数を解答に追加する

= 2 (\frac{3}{4} 4 x^{2} - 12 x + 6 - \frac{1}{4} ln \frac{4 x ^{2} - 12 x + 6}{3} + \frac{2}{3} x - 3) + C

グラフ

人気の例

xyy^'=sqrt(1-y^2)

x y y^{^{'}} = 1 - y^{2}

integral of r^2sqrt(16-r^2)

\int r^{2} 16 - r^{2} d r

derivative of (x^2(x^2+5)^2)

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

derivative of 200x

\frac{d}{d x} (200 x)

derivative of (x^{-1/2}+2x(7-x^{-1}))

\frac{d}{d x} ((x^{- \frac{1}{2}} + 2 x) (7 - x^{- 1}))