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受欢迎的微积分 >

integral of 1/(2sqrt(x)+3)

解答

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

解答

x + \frac{3}{2} - \frac{3}{2} ln 2 x + 3 + C

求解步骤

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

使用换元积分法

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

提出常数:

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

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

使用换元积分法

= 2 \cdot \int \frac{v - 3}{4 v} d v

提出常数:

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

= 2 \cdot \frac{1}{4} \cdot \int \frac{v - 3}{v} d v

乘开

\frac{v - 3}{v} : 1 - \frac{3}{v}

= 2 \cdot \frac{1}{4} \cdot \int 1 - \frac{3}{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

= 2 \cdot \frac{1}{4} (\int 1 d v - \int \frac{3}{v} d v)

\int 1 d v = v

\int \frac{3}{v} d v = 3 ln ∣ v ∣

= 2 \cdot \frac{1}{4} (v - 3 ln ∣ v ∣)

代回

= 2 \cdot \frac{1}{4} (2 x + 3 - 3 ln 2 x + 3)

化简

2 \cdot \frac{1}{4} (2 x + 3 - 3 ln 2 x + 3) : x + \frac{3}{2} - \frac{3}{2} ln 2 x + 3

= x + \frac{3}{2} - \frac{3}{2} ln 2 x + 3

解答补常数

= x + \frac{3}{2} - \frac{3}{2} ln 2 x + 3 + C

作图

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