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 cos^4(2y)

解答

\int cos^{4} (2 y) d y

解答

\frac{1}{2} (cos^{3} (2 y) sin (2 y) + \frac{3}{8} (2 y - \frac{1}{4} sin (8 y))) + C

求解步骤

\int cos^{4} (2 y) d y

使用换元积分法

= \int cos^{4} (u) \frac{1}{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 cos^{4} (u) d u

化简

cos^{4} (u) : cos^{3} (u) cos (u)

= \frac{1}{2} \cdot \int cos^{3} (u) cos (u) d u

使用分布积分法

= \frac{1}{2} (cos^{3} (u) sin (u) - \int - 3 cos^{2} (u) sin^{2} (u) d u)

\int - 3 cos^{2} (u) sin^{2} (u) d u = - \frac{3}{8} (u - \frac{1}{4} sin (4 u))

= \frac{1}{2} (cos^{3} (u) sin (u) - (- \frac{3}{8} (u - \frac{1}{4} sin (4 u))))

u = 2 y

代回

= \frac{1}{2} (cos^{3} (2 y) sin (2 y) - (- \frac{3}{8} (2 y - \frac{1}{4} sin (4 \cdot 2 y))))

化简

= \frac{1}{2} (cos^{3} (2 y) sin (2 y) + \frac{3}{8} (2 y - \frac{1}{4} sin (8 y)))

解答补常数

= \frac{1}{2} (cos^{3} (2 y) sin (2 y) + \frac{3}{8} (2 y - \frac{1}{4} sin (8 y))) + C

作图

流行的例子

(\partial)/(\partial x)(e^{-2x}cos(2pit))

\frac{\partial}{\partial x} (e^{- 2 x} cos (2 π t))

y^'=((4-2x))/((3+2y))

y^{^{'}} = \frac{( 4 - 2 x )}{( 3 + 2 y )}

tangent (3x)/(x^2-6)

t an g e n t \frac{3 x}{x ^{2} - 6}

derivative y=5+5x^2-2x^3

d er i v a t i v e y = 5 + 5 x^{2} - 2 x^{3}

derivative of e^x(1+x)

\frac{d}{d x} (e^{x} (1 + x))