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

解答

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

解答

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

求解步骤

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

使用换元积分法

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

使用换元积分法

= \int \frac{1}{3 ( 1 - v ^{2} )} d v

提出常数:

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

= \frac{1}{3} \cdot \int \frac{1}{1 - v ^{2}} d v

使用积分常见定则:

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

= \frac{1}{3} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

代回

= \frac{1}{3} \frac{ln \frac{x - 2}{3} + 1}{2} - \frac{ln \frac{x - 2}{3} - 1}{2}

化简

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

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

解答补常数

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

作图

流行的例子

integral of 4-|x|

\int 4 - ∣ x ∣ d x

integral from-1 to 1 of sqrt(x+1)-1

\int_{- 1}^{1} x + 1 - 1 d x

integral from-3 to 2 of |x+1|

\int_{- 3}^{2} ∣ x + 1 ∣ d x

limit as x approaches 1 of (2x)/(ln(x))

x \to 1 lim (\frac{2 x}{ln ( x )})

derivative f(x)=sqrt(6-x)

d er i v a t i v e f (x) = 6 - x