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(5+4x-x^2))

解答

\int \frac{x}{5 + 4 x - x ^{2}} d x

解答

- - x^{2} + 4 x + 5 + 2 arcsin (\frac{1}{3} (x - 2)) + C

求解步骤

\int \frac{x}{5 + 4 x - x ^{2}} d x

对

5 + 4 x - x^{2}

配方:

- (x - 2)^{2} + 9

= \int \frac{x}{- ( x - 2 ) ^{2} + 9} d x

使用换元积分法

= \int \frac{u + 2}{- u ^{2} + 9} d u

使用三角换元法

= \int 3 sin (v) + 2 d v

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 3 sin (v) d v + \int 2 d v

\int 3 sin (v) d v = - 3 cos (v)

\int 2 d v = 2 v

= - 3 cos (v) + 2 v

代回

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

化简

- 3 cos (arcsin (\frac{1}{3} (x - 2))) + 2 arcsin (\frac{1}{3} (x - 2)) : - - x^{2} + 4 x + 5 + 2 arcsin (\frac{1}{3} (x - 2))

= - - x^{2} + 4 x + 5 + 2 arcsin (\frac{1}{3} (x - 2))

解答补常数

= - - x^{2} + 4 x + 5 + 2 arcsin (\frac{1}{3} (x - 2)) + C

作图

流行的例子

integral from 1 to infinity of x^2

\int_{1}^{\infty} x^{2} d x

derivative f(x)=(2x^2+6x+6)/(x+1)

d er i v a t i v e f (x) = \frac{2 x ^{2} + 6 x + 6}{x + 1}

(dy)/(dx)=e^{4y}sin(5x)

\frac{d y}{d x} = e^{4 y} sin (5 x)

(\partial)/(\partial x)(6x+4y)

\frac{\partial}{\partial x} (6 x + 4 y)

integral of (x^4)^{1/2}

\int (x^{4})^{\frac{1}{2}} d x