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+2x-1)

解答

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

解答

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

求解步骤

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

因式分解

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

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

对

- x^{2} - 2 x + 1

配方:

- (x + 1)^{2} + 2

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

使用换元积分法

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

乘开

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

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

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

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

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

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

u = x + 1

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

integral of (2x^5-x^2+1)/(x^4)

\int \frac{2 x ^{5} - x ^{2} + 1}{x ^{4}} d x

integral of ln^2(x)sqrt(x)

\int ln^{2} (x) x d x

integral of (6x^2+1)/(8x)

\int \frac{6 x ^{2} + 1}{8 x} d x

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

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

integral of 2x^3-3x^2

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