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 (e^{2x})/(\sqrt[3]{1+e^x)}

解答

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

解答

\frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}} + C

求解步骤

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

使用换元积分法

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

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

使用换元积分法

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

乘开

\frac{w - 1}{3 w} : w^{\frac{2}{3}} - \frac{1}{3 w}

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

\int w^{\frac{2}{3}} d w = \frac{3}{5} w^{\frac{5}{3}}

\int \frac{1}{3 w} d w = \frac{3}{2} w^{\frac{2}{3}}

= \frac{1}{2} \cdot 2 (\frac{3}{5} w^{\frac{5}{3}} - \frac{3}{2} w^{\frac{2}{3}})

代回

= \frac{1}{2} \cdot 2 (\frac{3}{5} (1 + e^{2 x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{2 x})^{\frac{2}{3}})

化简

\frac{1}{2} \cdot 2 (\frac{3}{5} (1 + e^{2 x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{2 x})^{\frac{2}{3}}) : \frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}}

= \frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}}

解答补常数

= \frac{3}{5} (1 + e^{x})^{\frac{5}{3}} - \frac{3}{2} (1 + e^{x})^{\frac{2}{3}} + C

作图

流行的例子

integral of (3x-6)^2

\int (3 x - 6)^{2} d x

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

\int \frac{x - 2}{x ^{2} - 2 x + 2} d x

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

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

integral of (2cos(x)+3sin(x))/(sin^3(x))

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

integral of 2sqrt(x)(x+2x^{1/3})

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