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(1-x^4)

解答

\int x \cdot 1 - x^{4} d x

解答

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

求解步骤

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

化简

x^{4} : (x^{2})^{2}

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

使用换元积分法

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

使用三角换元法

= \frac{1}{2} \cdot \int cos^{2} (v) d v

使用三角恒等式改写

= \frac{1}{2} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

提出常数:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) 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

= \frac{1}{2} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= \frac{1}{2} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

代回

= \frac{1}{2} \cdot \frac{1}{2} (arcsin (x^{2}) + \frac{1}{2} sin (2 arcsin (x^{2})))

化简

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

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

解答补常数

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

作图

流行的例子

sum from n=0 to infinity of (1/2)^{n+1}

n = 0 \sum \infty (\frac{1}{2})^{n + 1}

y^{^{'}} + \frac{y}{x + 1} = 0

derivative y=(8x^2+2x+8)/(sqrt(x))

d er i v a t i v e y = \frac{8 x ^{2} + 2 x + 8}{x}

derivative x/(x^2+2)

d er i v a t i v e \frac{x}{x ^{2} + 2}

derivative of (e^{2x}-e^x+e^{-3x}/(e^x))

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