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

解答

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

解答

- 2 - x^{2} + 4 x + arcsin (\frac{1}{2} x - 1) + C

求解步骤

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

对

4 x - x^{2}

配方:

- (x - 2)^{2} + 4

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

使用换元积分法

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

乘开

\frac{2 u + 1}{- u ^{2} + 4} : \frac{2 u}{- u ^{2} + 4} + \frac{1}{- u ^{2} + 4}

= \int \frac{2 u}{- u ^{2} + 4} + \frac{1}{- u ^{2} + 4} 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

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

\int \frac{2 u}{- u ^{2} + 4} d u = - 2 - u^{2} + 4

\int \frac{1}{- u ^{2} + 4} d u = arcsin (\frac{1}{2} u)

= - 2 - u^{2} + 4 + arcsin (\frac{1}{2} u)

u = x - 2

代回

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

化简

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

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

解答补常数

= - 2 - x^{2} + 4 x + arcsin (\frac{1}{2} x - 1) + C

作图

流行的例子

integral from e to e^2 of 1/(xln(x))

\int_{e}^{e^{2}} \frac{1}{x ln ( x )} d x

derivative of (40/(1+79e^{-x)})

\frac{d}{d x} (\frac{40}{1 + 79 e ^{- x}})

derivative of x/(sqrt(x-7))

\frac{d}{d x} (\frac{x}{x - 7})

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

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

(\partial)/(\partial y)(5xy^2)

\frac{\partial}{\partial y} (5 x y^{2})