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 sin^4(2x)*cos^2(2x)

解答

\int sin^{4} (2 x) \cdot cos^{2} (2 x) d x

解答

\frac{1}{192} (12 x - 3 sin (4 x) - 4 sin^{3} (2 x) cos (2 x) + 16 sin^{5} (2 x) cos (2 x)) + C

求解步骤

\int sin^{4} (2 x) cos^{2} (2 x) d x

使用换元积分法

= \int sin^{4} (u) cos^{2} (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 sin^{4} (u) cos^{2} (u) d u

使用三角恒等式改写

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

乘开

sin^{4} (u) (1 - sin^{2} (u)) : sin^{4} (u) - sin^{6} (u)

= \frac{1}{2} \cdot \int sin^{4} (u) - sin^{6} (u) d u

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} (\int sin^{4} (u) d u - \int sin^{6} (u) d u)

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

\int sin^{6} (u) d u = - \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)))

= \frac{1}{2} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)) - (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u)))))

u = 2 x

代回

= \frac{1}{2} (- \frac{1}{4} sin^{3} (2 x) cos (2 x) + \frac{3}{8} (2 x - \frac{1}{2} sin (2 \cdot 2 x)) - (- \frac{cos ( 2 x ) sin ^{5} ( 2 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (2 x) cos (2 x) + \frac{3}{8} (2 x - \frac{1}{2} sin (2 \cdot 2 x)))))

化简

\frac{1}{2} (- \frac{1}{4} sin^{3} (2 x) cos (2 x) + \frac{3}{8} (2 x - \frac{1}{2} sin (2 \cdot 2 x)) - (- \frac{cos ( 2 x ) sin ^{5} ( 2 x )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (2 x) cos (2 x) + \frac{3}{8} (2 x - \frac{1}{2} sin (2 \cdot 2 x))))) : \frac{1}{192} (12 x - 3 sin (4 x) - 4 sin^{3} (2 x) cos (2 x) + 16 sin^{5} (2 x) cos (2 x))

= \frac{1}{192} (12 x - 3 sin (4 x) - 4 sin^{3} (2 x) cos (2 x) + 16 sin^{5} (2 x) cos (2 x))

解答补常数

= \frac{1}{192} (12 x - 3 sin (4 x) - 4 sin^{3} (2 x) cos (2 x) + 16 sin^{5} (2 x) cos (2 x)) + C

作图

流行的例子

limit as x approaches 8-of (|x+8|)/(x+8)

x \to 8 - lim (\frac{∣ x + 8 ∣}{x + 8})

integral of sin^{(2)}(x)

\int sin^{(2)} (x) d x

limit as x approaches 16 of sqrt(x+x^2)

x \to 16 lim (x + x^{2})

inverselaplace 4+t^6

in v erse l a pl a ce 4 + t^{6}

limit as x approaches-2 of-x^2+9x-4

x \to - 2 lim (- x^{2} + 9 x - 4)