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受欢迎的微积分 >

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

解答

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

解答

6 (x + 2 - 2 ln x + 2) + C

求解步骤

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

提出常数:

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

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

使用换元积分法

= 3 \cdot \int \frac{2 ( u - 2 )}{u} d u

提出常数:

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

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

乘开

\frac{u - 2}{u} : 1 - \frac{2}{u}

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

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

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

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= 3 \cdot 2 (u - 2 ln ∣ u ∣)

u = x + 2

代回

= 3 \cdot 2 (x + 2 - 2 ln x + 2)

化简

= 6 (x + 2 - 2 ln x + 2)

解答补常数

= 6 (x + 2 - 2 ln x + 2) + C

作图

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