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 from 0 to 1 of pi(x-x^4)^2

解答

\int_{0}^{1} π (x - x^{4})^{2} d x

解答

\frac{π}{9}

+1

十进制

0.34906 \dots

求解步骤

\int_{0}^{1} π (x - x^{4})^{2} d x

提出常数:

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

= π \cdot \int_{0}^{1} (x - x^{4})^{2} d x

因式分解

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

= π \cdot \int_{0}^{1} (- (- x + x^{4}))^{2} d x

化简

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

= π \cdot \int_{0}^{1} (- x + x^{4})^{2} d x

乘开

(- x + x^{4})^{2} : x^{2} - 2 x^{5} + x^{8}

= π \cdot \int_{0}^{1} x^{2} - 2 x^{5} + x^{8} d x

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

= π (\int_{0}^{1} x^{2} d x - \int_{0}^{1} 2 x^{5} d x + \int_{0}^{1} x^{8} d x)

\int_{0}^{1} x^{2} d x = \frac{1}{3}

\int_{0}^{1} 2 x^{5} d x = \frac{1}{3}

\int_{0}^{1} x^{8} d x = \frac{1}{9}

= π (\frac{1}{3} - \frac{1}{3} + \frac{1}{9})

化简

π (\frac{1}{3} - \frac{1}{3} + \frac{1}{9}) : \frac{π}{9}

= \frac{π}{9}

作图

流行的例子

laplacetransform e^{t+7}

l a pl a ce t r an s f or m e^{t + 7}

integral of 5xsqrt(1-x^4)

\int 5 x 1 - x^{4} d x

derivative x(x^2+4)^5

d er i v a t i v e x (x^{2} + 4)^{5}

derivative f(x)=5x+2x^2+7e^x

d er i v a t i v e f (x) = 5 x + 2 x^{2} + 7 e^{x}

面积 y=-x^2+1,y=0

a re a y = - x^{2} + 1, y = 0