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^3+3x)/(x^2+1)

解答

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

解答

\frac{1}{2} x^{2} + \frac{1}{2} + ln x^{2} + 1 + C

求解步骤

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

使用换元积分法

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

提出常数:

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

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

乘开

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

= \frac{1}{2} \cdot \int 1 + \frac{2}{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

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

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{1}{2} (u + 2 ln ∣ u ∣)

u = x^{2} + 1

代回

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

化简

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

= \frac{1}{2} x^{2} + \frac{1}{2} + ln x^{2} + 1

解答补常数

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

作图

流行的例子

integral of ((ln(x^6)))/x

\int \frac{( ln ( x ^{6} ) )}{x} d x

derivative f(x)=(2-3x^3)^{-2}

d er i v a t i v e f (x) = (2 - 3 x^{3})^{- 2}

integral from 0 to pi of 9sin^4(3t)

\int_{0}^{π} 9 sin^{4} (3 t) d t

t an g e n t 6 - 2 e^{x}

limit as x approaches 0 of 2cos(pi/x)

x \to 0 lim (2 cos (\frac{π}{x}))