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^5(7x)

解答

\int sin^{5} (7 x) d x

解答

\frac{1}{7} (- cos (7 x) + \frac{2 cos ^{3} ( 7 x )}{3} - \frac{cos ^{5} ( 7 x )}{5}) + C

求解步骤

\int sin^{5} (7 x) d x

使用换元积分法

= \int sin^{5} (u) \frac{1}{7} d u

提出常数:

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

= \frac{1}{7} \cdot \int sin^{5} (u) d u

使用三角恒等式改写

= \frac{1}{7} \cdot \int (1 - cos^{2} (u))^{2} sin (u) d u

使用换元积分法

= \frac{1}{7} \cdot \int - 1 + 2 v^{2} - v^{4} 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}{7} (- \int 1 d v + \int 2 v^{2} d v - \int v^{4} d v)

\int 1 d v = v

\int 2 v^{2} d v = \frac{2 v ^{3}}{3}

\int v^{4} d v = \frac{v ^{5}}{5}

= \frac{1}{7} (- v + \frac{2 v ^{3}}{3} - \frac{v ^{5}}{5})

代回

= \frac{1}{7} (- cos (7 x) + \frac{2 cos ^{3} ( 7 x )}{3} - \frac{cos ^{5} ( 7 x )}{5})

解答补常数

= \frac{1}{7} (- cos (7 x) + \frac{2 cos ^{3} ( 7 x )}{3} - \frac{cos ^{5} ( 7 x )}{5}) + C

作图

流行的例子

(dy)/(dx)=32x^3y^3

\frac{d y}{d x} = 32 x^{3} y^{3}

limit as x approaches 0 of (1-1/(2x))^x

x \to 0 lim ((1 - \frac{1}{2 x})^{x})

d/(dy)(ye^x+2e^x+y^2)

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

integral of sqrt((2x+1)^5)

\int (2 x + 1)^{5} d x

inverselaplace (-2)/((s^2+1)^2)

in v erse l a pl a ce \frac{- 2}{( s ^{2} + 1 ) ^{2}}