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^2)/((1+2x)^{1/3)}

解答

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

解答

\frac{1}{320} (60 x^{2} (1 + 2 x)^{\frac{2}{3}} - 36 x (1 + 2 x)^{\frac{2}{3}} + 27 (1 + 2 x)^{\frac{2}{3}}) + C

求解步骤

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

使用指数法则:

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

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

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

使用换元积分法

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

提出常数:

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

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

使用分布积分法

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

\int \frac{5 u ^{2} - 4 u - 1}{3 u ^{\frac{1}{3}}} d u = \frac{1}{40} u^{\frac{2}{3}} (25 u^{2} - 32 u - 20)

= \frac{1}{8} (u^{\frac{2}{3}} (u - 1)^{2} - \frac{1}{40} u^{\frac{2}{3}} (25 u^{2} - 32 u - 20))

u = 1 + 2 x

代回

= \frac{1}{8} ((1 + 2 x)^{\frac{2}{3}} (1 + 2 x - 1)^{2} - \frac{1}{40} (1 + 2 x)^{\frac{2}{3}} (25 (1 + 2 x)^{2} - 32 (1 + 2 x) - 20))

化简

\frac{1}{8} ((1 + 2 x)^{\frac{2}{3}} (1 + 2 x - 1)^{2} - \frac{1}{40} (1 + 2 x)^{\frac{2}{3}} (25 (1 + 2 x)^{2} - 32 (1 + 2 x) - 20)) : \frac{1}{320} (60 x^{2} (1 + 2 x)^{\frac{2}{3}} - 36 x (1 + 2 x)^{\frac{2}{3}} + 27 (1 + 2 x)^{\frac{2}{3}})

= \frac{1}{320} (60 x^{2} (1 + 2 x)^{\frac{2}{3}} - 36 x (1 + 2 x)^{\frac{2}{3}} + 27 (1 + 2 x)^{\frac{2}{3}})

解答补常数

= \frac{1}{320} (60 x^{2} (1 + 2 x)^{\frac{2}{3}} - 36 x (1 + 2 x)^{\frac{2}{3}} + 27 (1 + 2 x)^{\frac{2}{3}}) + C

作图

流行的例子

(\partial)/(\partial y)(e^{xy}y^2+x/y)

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

(\partial)/(\partial r)(sqrt(r^2+s^2))

\frac{\partial}{\partial r} (r^{2} + s^{2})

derivative of sqrt(sin(\sqrt{x)})

\frac{d}{d x} (sin (x))

derivative y=(7-3x^3)^3

d er i v a t i v e y = (7 - 3 x^{3})^{3}

integral of 16e^x

\int 16 e^{x} d x