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

解答

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

解答

\frac{9}{2} (1 + x^{\frac{2}{3}} - ln 1 + x^{\frac{2}{3}}) + C

求解步骤

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

提出常数:

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

= 3 \cdot \int \frac{x ^{\frac{1}{3}}}{1 + x ^{\frac{2}{3}}} d x

使用换元积分法

= 3 \cdot \int \frac{3 u ^{3}}{1 + u ^{2}} d u

提出常数:

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

= 3 \cdot 3 \cdot \int \frac{u ^{3}}{1 + u ^{2}} d u

使用换元积分法

= 3 \cdot 3 \cdot \int \frac{v - 1}{2 v} d v

提出常数:

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

= 3 \cdot 3 \cdot \frac{1}{2} \cdot \int \frac{v - 1}{v} d v

乘开

\frac{v - 1}{v} : 1 - \frac{1}{v}

= 3 \cdot 3 \cdot \frac{1}{2} \cdot \int 1 - \frac{1}{v} 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

= 3 \cdot 3 \cdot \frac{1}{2} (\int 1 d v - \int \frac{1}{v} d v)

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= 3 \cdot 3 \cdot \frac{1}{2} (v - ln ∣ v ∣)

代回

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

化简

3 \cdot 3 \cdot \frac{1}{2} (1 + (x^{\frac{1}{3}})^{2} - ln 1 + (x^{\frac{1}{3}})^{2}) : \frac{9}{2} (1 + x^{\frac{2}{3}} - ln 1 + x^{\frac{2}{3}})

= \frac{9}{2} (1 + x^{\frac{2}{3}} - ln 1 + x^{\frac{2}{3}})

解答补常数

= \frac{9}{2} (1 + x^{\frac{2}{3}} - ln 1 + x^{\frac{2}{3}}) + C

作图

流行的例子

integral of (4x^2+2+5/(x^2+1))

\int (4 x^{2} + 2 + \frac{5}{x ^{2} + 1}) d x

integral of (e^{2x}+2e^{3x}+2)e^x

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

integral of x^4sqrt(8x^5-7)

\int x^{4} 8 x^{5} - 7 d x

integral of sin(12x)cos^4(12x)

\int sin (12 x) cos^{4} (12 x) d x

integral of 1/(sqrt(5s+6))

\int \frac{1}{5 s + 6} d s