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^3(xco)s^{-2}x

解答

\int sin^{3} (x co) s^{- 2} x d x

解答

\frac{1}{c ^{2} o ^{2} s ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x)) + C

求解步骤

\int sin^{3} (x co) s^{- 2} x d x

提出常数:

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

= s^{- 2} \cdot \int sin^{3} (co x) x d x

使用指数法则:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{s ^{2}} \cdot \int sin^{3} (co x) x d x

使用换元积分法

= \frac{1}{s ^{2}} \cdot \int \frac{u sin ^{3} ( u )}{c ^{2} o ^{2}} d u

提出常数:

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

= \frac{1}{s ^{2}} \cdot \frac{1}{c ^{2} o ^{2}} \cdot \int u sin^{3} (u) d u

使用分布积分法

= \frac{1}{s ^{2}} \cdot \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + \frac{1}{3} cos^{3} (u)) - \int - cos (u) + \frac{1}{3} cos^{3} (u) d u)

\int - cos (u) + \frac{1}{3} cos^{3} (u) d u = - \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u)

= \frac{1}{s ^{2}} \cdot \frac{1}{c ^{2} o ^{2}} (u (- cos (u) + \frac{1}{3} cos^{3} (u)) - (- \frac{2}{3} sin (u) - \frac{1}{9} sin^{3} (u)))

u = co x

代回

= \frac{1}{s ^{2}} \cdot \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + \frac{1}{3} cos^{3} (co x)) - (- \frac{2}{3} sin (co x) - \frac{1}{9} sin^{3} (co x)))

化简

\frac{1}{s ^{2}} \cdot \frac{1}{c ^{2} o ^{2}} (co x (- cos (co x) + \frac{1}{3} cos^{3} (co x)) - (- \frac{2}{3} sin (co x) - \frac{1}{9} sin^{3} (co x))) : \frac{1}{c ^{2} o ^{2} s ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x))

= \frac{1}{c ^{2} o ^{2} s ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x))

解答补常数

= \frac{1}{c ^{2} o ^{2} s ^{2}} (- co x cos (co x) + \frac{1}{3} co x cos^{3} (co x) + \frac{2}{3} sin (co x) + \frac{1}{9} sin^{3} (co x)) + C

作图

流行的例子

derivative sqrt(7-x)

d er i v a t i v e 7 - x

derivative sin^5(x)

d er i v a t i v e sin^{5} (x)

integral of (x+3)/(sqrt(4-(x-2)^2))

\int \frac{x + 3}{4 - ( x - 2 ) ^{2}} d x

derivative 2^{90}

d er i v a t i v e 2^{90}

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

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