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 2/(2+cos(x))

解答

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

解答

\frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + C

求解步骤

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

提出常数:

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

= 2 \cdot \int \frac{1}{2 + cos ( x )} d x

使用换元积分法

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

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

= 2 \cdot 2 \cdot \frac{1}{3} arctan (v)

代回

= 2 \cdot 2 \cdot \frac{1}{3} arctan (\frac{tan ( \frac{x}{2} )}{3})

化简

2 \cdot 2 \cdot \frac{1}{3} arctan (\frac{tan ( \frac{x}{2} )}{3}) : \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2}))

= \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2}))

解答补常数

= \frac{4}{3} arctan (\frac{1}{3} tan (\frac{x}{2})) + C

作图

流行的例子

integral of 1/(s^2+1)

\int \frac{1}{s ^{2} + 1} d s

(\partial)/(\partial x)(e^{9x+5y})

\frac{\partial}{\partial x} (e^{9 x + 5 y})

derivative of (2x^2+1/((1-x^2)^{5/2)})

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

integral of 1/(2*sqrt(tan(x)))

\int \frac{1}{2 \cdot tan ( x )} d x

integral of x/(x+\sqrt[3]{x)}

\int \frac{x}{x + 3 x} d x