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 2pi of 6x^2sin(x/4)

解答

\int_{0}^{2 π} 6 x^{2} sin (\frac{x}{4}) d x

解答

384 (π - 2)

+1

十进制

438.37157 \dots

求解步骤

\int_{0}^{2 π} 6 x^{2} sin (\frac{x}{4}) d x

提出常数:

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

= 6 \cdot \int_{0}^{2 π} x^{2} sin (\frac{x}{4}) d x

使用换元积分法

= 6 \cdot \int_{0}^{\frac{π}{2}} 64 u^{2} sin (u) d u

提出常数:

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

= 6 \cdot 64 \cdot \int_{0}^{\frac{π}{2}} u^{2} sin (u) d u

使用分布积分法

= 6 \cdot 64 [- u^{2} cos (u) - \int - 2 u cos (u) d u]_{0}^{\frac{π}{2}}

\int - 2 u cos (u) d u = - 2 (u sin (u) + cos (u))

= 6 \cdot 64 [- u^{2} cos (u) - (- 2 (u sin (u) + cos (u)))]_{0}^{\frac{π}{2}}

化简

= 384 [- u^{2} cos (u) + 2 (u sin (u) + cos (u))]_{0}^{\frac{π}{2}}

带入上下限计算后得:

π - 2

= 384 (π - 2)

作图

流行的例子

derivative xsin(2/x)

d er i v a t i v e x sin (\frac{2}{x})

integral of (3x)/(sqrt(4-x))

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

integral of ((x^2-x+12))/(x^3+2x)

\int \frac{( x ^{2} - x + 12 )}{x ^{3} + 2 x} d x

limit as t approaches 0 of ((3e^t-3))/t

t \to 0 lim (\frac{( 3 e ^{t} - 3 )}{t})

derivative (f(x))^3

d er i v a t i v e (f (x))^{3}