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受欢迎的微积分 >

integral of 1/(2+cos(u))

解答

\int \frac{1}{2 + cos ( u )} d u

解答

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

求解步骤

\int \frac{1}{2 + cos ( u )} d u

使用换元积分法

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

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

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

代回

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

化简

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

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

解答补常数

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

作图

流行的例子

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

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

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