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

解答

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

解答

- - x^{2} - 2 x + 4 - arcsin (\frac{1}{5} (x + 1)) + C

求解步骤

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

对

4 - 2 x - x^{2}

配方:

- (x + 1)^{2} + 5

= \int \frac{x}{- ( x + 1 ) ^{2} + 5} d x

使用换元积分法

= \int \frac{u - 1}{- u ^{2} + 5} d u

乘开

\frac{u - 1}{- u ^{2} + 5} : \frac{u}{- u ^{2} + 5} - \frac{1}{- u ^{2} + 5}

= \int \frac{u}{- u ^{2} + 5} - \frac{1}{- u ^{2} + 5} 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{u}{- u ^{2} + 5} d u - \int \frac{1}{- u ^{2} + 5} d u

\int \frac{u}{- u ^{2} + 5} d u = - - u^{2} + 5

\int \frac{1}{- u ^{2} + 5} d u = arcsin (\frac{1}{5} u)

= - - u^{2} + 5 - arcsin (\frac{1}{5} u)

u = x + 1

代回

= - - (x + 1)^{2} + 5 - arcsin (\frac{1}{5} (x + 1))

乘开

- (x + 1)^{2} + 5 : - x^{2} - 2 x + 4

= - - x^{2} - 2 x + 4 - arcsin (\frac{1}{5} (x + 1))

解答补常数

= - - x^{2} - 2 x + 4 - arcsin (\frac{1}{5} (x + 1)) + C

作图

流行的例子

integral of 5/(x^{3/2)-25sqrt(x)}

\int \frac{5}{x ^{\frac{3}{2}} - 25 x} d x

integral of sqrt(7-4x)

\int 7 - 4 x d x

integral of 240sin(2x)

\int 240 sin (2 x) d x

integral of sin(7x)cos(6x)

\int sin (7 x) cos (6 x) d x

integral of ((x^3-x)^2)/x

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