zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of (-x)/(x^2-x+1)

解答

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

解答

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

求解步骤

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

提出常数:

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

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

对

x^{2} - x + 1

配方:

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

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

使用换元积分法

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

提出常数:

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

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

乘开

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

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 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)

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

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

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

integral of x/(sqrt(25x^4+9))

\int \frac{x}{25 x ^{4} + 9} d x

integral of 2^x*3^x

\int 2^{x} \cdot 3^{x} d x

integral of 2sin(2x)-3e^x

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

integral of (x^2+3)/(xsqrt(x^2-4))

\int \frac{x ^{2} + 3}{x x ^{2} - 4} d x

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

\int x^{3} + x - \frac{1}{x ^{2}} d x