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 x^5arccos(sqrt(1-x^6))

解答

\int x^{5} arccos (1 - x^{6}) d x

解答

- \frac{1}{3} (\frac{1}{2} arccos (1 - x^{6}) (1 - x^{6}) + \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6})))) + C

求解步骤

\int x^{5} arccos (1 - x^{6}) d x

使用换元积分法

= \int - \frac{arccos ( u )}{6} d u

提出常数:

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

= - \frac{1}{6} \cdot \int arccos (u) d u

使用换元积分法

= - \frac{1}{6} \cdot \int arccos (v) \cdot 2 v d v

提出常数:

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

= - \frac{1}{6} \cdot 2 \cdot \int arccos (v) v d v

使用分布积分法

= - \frac{1}{6} \cdot 2 (\frac{1}{2} v^{2} arccos (v) - \int - \frac{v ^{2}}{2 1 - v ^{2}} d v)

\int - \frac{v ^{2}}{2 1 - v ^{2}} d v = - \frac{1}{4} (arcsin (v) - \frac{1}{2} sin (2 arcsin (v)))

= - \frac{1}{6} \cdot 2 (\frac{1}{2} v^{2} arccos (v) - (- \frac{1}{4} (arcsin (v) - \frac{1}{2} sin (2 arcsin (v)))))

代回

= - \frac{1}{6} \cdot 2 (\frac{1}{2} (1 - x^{6})^{2} arccos (1 - x^{6}) - (- \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6})))))

化简

- \frac{1}{6} \cdot 2 (\frac{1}{2} (1 - x^{6})^{2} arccos (1 - x^{6}) - (- \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6}))))) : - \frac{1}{3} (\frac{1}{2} arccos (1 - x^{6}) (1 - x^{6}) + \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6}))))

= - \frac{1}{3} (\frac{1}{2} arccos (1 - x^{6}) (1 - x^{6}) + \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6}))))

解答补常数

= - \frac{1}{3} (\frac{1}{2} arccos (1 - x^{6}) (1 - x^{6}) + \frac{1}{4} (arcsin (1 - x^{6}) - \frac{1}{2} sin (2 arcsin (1 - x^{6})))) + C

作图

流行的例子

integral of x/((x^2-2))

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

integral of-6cos(t)

\int - 6 cos (t) d t

integral of x/((2x-1)^{1/2)}

\int \frac{x}{( 2 x - 1 ) ^{\frac{1}{2}}} d x

integral of (1-4x)^{2/3}

\int (1 - 4 x)^{\frac{2}{3}} d x

integral of-8csc^2(x)

\int - 8 csc^{2} (x) d x