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+2)/(x^2+2x+3)

解答

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

解答

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

求解步骤

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

对

x^{2} + 2 x + 3

配方:

(x + 1)^{2} + 2

= \int \frac{x + 2}{( 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} arctan (\frac{u}{2})

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

u = x + 1

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

integral of (e^x)/((e^x+1)^{13)}

\int \frac{e ^{x}}{( e ^{x} + 1 ) ^{13}} d x

integral of (3x^{b-1})/(a+x^b)

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

integral of ln(xe^x)

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

integral of (x^2)/(sqrt(1-4x^2))

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

integral of (12x^{3/4}-9x^{5/3})

\int (12 x^{\frac{3}{4}} - 9 x^{\frac{5}{3}}) d x