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 ((2x^3)/(x^2-4))

解答

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

解答

x^{2} - 4 + 4 ln x^{2} - 4 + C

求解步骤

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

提出常数:

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

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

使用换元积分法

= 2 \cdot \int \frac{u + 4}{2 u} d u

提出常数:

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

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

乘开

\frac{u + 4}{u} : 1 + \frac{4}{u}

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

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 \cdot \frac{1}{2} (\int 1 d u + \int \frac{4}{u} d u)

\int 1 d u = u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= 2 \cdot \frac{1}{2} (u + 4 ln ∣ u ∣)

u = x^{2} - 4

代回

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

化简

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

= x^{2} - 4 + 4 ln x^{2} - 4

解答补常数

= x^{2} - 4 + 4 ln x^{2} - 4 + C

作图

流行的例子

integral from 0 to pi of (cos(x))

\int_{0}^{π} (cos (x)) d x

limit as x approaches 1 of x/(3x+2)

x \to 1 lim (\frac{x}{3 x + 2})

derivative f(x)=sqrt(9-7x)

d er i v a t i v e f (x) = 9 - 7 x

derivative of e^{4x}cos(2x)

\frac{d}{d x} (e^{4 x} cos (2 x))

derivative of kx^x

\frac{d}{d x} (k x^{x})