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 (sqrt(x))/(x+x^{4/5)}

解答

\int \frac{x}{x + x ^{\frac{4}{5}}} d x

解答

10 (\frac{1}{5} x - \frac{1}{3} x^{\frac{3}{10}} + 10 x - arctan (10 x)) + C

求解步骤

\int \frac{x}{x + x ^{\frac{4}{5}}} d x

使用换元积分法

= \int \frac{10 u ^{6}}{u ^{2} + 1} d u

提出常数:

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

= 10 \cdot \int \frac{u ^{6}}{u ^{2} + 1} d u

长除

\frac{u ^{6}}{u ^{2} + 1} : u^{4} - u^{2} + 1 - \frac{1}{u ^{2} + 1}

= 10 \cdot \int u^{4} - u^{2} + 1 - \frac{1}{u ^{2} + 1} 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

= 10 (\int u^{4} d u - \int u^{2} d u + \int 1 d u - \int \frac{1}{u ^{2} + 1} d u)

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

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

\int 1 d u = u

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= 10 (\frac{u ^{5}}{5} - \frac{u ^{3}}{3} + u - arctan (u))

u = x^{\frac{1}{10}}

代回

= 10 \frac{( x ^{\frac{1}{10}} ) ^{5}}{5} - \frac{( x ^{\frac{1}{10}} ) ^{3}}{3} + x^{\frac{1}{10}} - arctan (x^{\frac{1}{10}})

化简

10 \frac{( x ^{\frac{1}{10}} ) ^{5}}{5} - \frac{( x ^{\frac{1}{10}} ) ^{3}}{3} + x^{\frac{1}{10}} - arctan (x^{\frac{1}{10}}) : 10 (\frac{1}{5} x - \frac{1}{3} x^{\frac{3}{10}} + 10 x - arctan (10 x))

= 10 (\frac{1}{5} x - \frac{1}{3} x^{\frac{3}{10}} + 10 x - arctan (10 x))

解答补常数

= 10 (\frac{1}{5} x - \frac{1}{3} x^{\frac{3}{10}} + 10 x - arctan (10 x)) + C

作图

流行的例子

integral from 3 to 6 of 8

\int_{3}^{6} 8 d x

integral of-5pisin(pix)

\int - 5 π sin (π x) d x

integral of 1/((1+x)^3)

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

derivative-2/(x^3)

d er i v a t i v e - \frac{2}{x ^{3}}

derivative of x/(sqrt(x-9))

\frac{d}{d x} (\frac{x}{x - 9})