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 1/(1+\sqrt[3]{ax-1)}

解答

\int \frac{1}{1 + 3 a x - 1} d x

解答

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

求解步骤

\int \frac{1}{1 + 3 a x - 1} d x

使用换元积分法

= \int \frac{1}{a ( 1 + 3 u )} d u

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

使用换元积分法

= \frac{1}{a} \cdot 3 \cdot \int \frac{( w - 1 ) ^{2}}{w} d w

乘开

\frac{( w - 1 ) ^{2}}{w} : w - 2 + \frac{1}{w}

= \frac{1}{a} \cdot 3 \cdot \int w - 2 + \frac{1}{w} d w

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

= \frac{1}{a} \cdot 3 (\int w d w - \int 2 d w + \int \frac{1}{w} d w)

\int w d w = \frac{w ^{2}}{2}

\int 2 d w = 2 w

\int \frac{1}{w} d w = ln ∣ w ∣

= \frac{1}{a} \cdot 3 (\frac{w ^{2}}{2} - 2 w + ln ∣ w ∣)

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

integral of 2x^2(x+2)^2

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

integral of x^2(x-4)

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

integral of 0.5Q

\int 0.5 Q d Q

integral of xsin(2x)cos(x)

\int x sin (2 x) cos (x) d x

integral of (e^{2x})/(3+e^{4x)}

\int \frac{e ^{2 x}}{3 + e ^{4 x}} d x