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 (8x)/(x^2+x+1)

解

\int \frac{8 x}{x ^{2} + x + 1} d x

解

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

解答ステップ

\int \frac{8 x}{x ^{2} + x + 1} d x

定数を除く:

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

= 8 \cdot \int \frac{x}{x ^{2} + x + 1} d x

平方完成する

x^{2} + x + 1 : (x + \frac{1}{2})^{2} + \frac{3}{4}

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

u置換積分法を適用する

= 8 \cdot \int \frac{2 ( 2 u - 1 )}{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

= 8 \cdot 2 \cdot \int \frac{2 u - 1}{4 u ^{2} + 3} d u

拡張

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

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

= 8 \cdot 2 (\int \frac{2 u}{4 u ^{2} + 3} d u - \int \frac{1}{4 u ^{2} + 3} d u)

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

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

= 8 \cdot 2 (\frac{1}{4} ln 4 u^{2} + 3 - \frac{1}{2 3} arctan (\frac{2}{3} u))

代用を戻す

u = x + \frac{1}{2}

= 8 \cdot 2 (\frac{1}{4} ln 4 (x + \frac{1}{2})^{2} + 3 - \frac{1}{2 3} arctan (\frac{2}{3} (x + \frac{1}{2})))

簡素化

8 \cdot 2 (\frac{1}{4} ln 4 (x + \frac{1}{2})^{2} + 3 - \frac{1}{2 3} arctan (\frac{2}{3} (x + \frac{1}{2}))) : 16 (\frac{1}{4} ln 4 x^{2} + 4 x + 4 - \frac{1}{2 3} arctan (\frac{2 x + 1}{3}))

= 16 (\frac{1}{4} ln 4 x^{2} + 4 x + 4 - \frac{1}{2 3} arctan (\frac{2 x + 1}{3}))

定数を解答に追加する

= 16 (\frac{1}{4} ln 4 x^{2} + 4 x + 4 - \frac{1}{2 3} arctan (\frac{2 x + 1}{3})) + C

グラフ

人気の例

limit as x approaches-2 of x^2-1

x \to - 2 lim (x^{2} - 1)

integral of (w+rs-1/c e^{(r-p)t})

\int (w + rs - \frac{1}{c} e^{(r - p) t}) d t

derivative of (sin(3x)^{ln(x)})

\frac{d}{d x} ((sin (3 x))^{l n (x)})

limit as n approaches infinity of 3/2

n \to \infty lim (\frac{3}{2})

(\partial)/(\partial x)(x/y)

\frac{\partial}{\partial x} (\frac{x}{y})