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受欢迎的微积分 >

integral of ((sqrt(3x)+4))/(sqrt(3x)-2)

解答

\int \frac{( 3 x + 4 )}{3 x - 2} d x

解答

\frac{1}{3} (3 x + 123 x + 24 ln 3 x - 2 - 28) + C

求解步骤

\int \frac{3 x + 4}{3 x - 2} d x

使用换元积分法

= \int \frac{u + 4}{3 ( u - 2 )} d u

提出常数:

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

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

乘开

\frac{u + 4}{u - 2} : \frac{u}{u - 2} + \frac{4}{u - 2}

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}{3} (\int \frac{u}{u - 2} d u + \int \frac{4}{u - 2} d u)

\int \frac{u}{u - 2} d u = u + 4 u + 8 ln u - 2 - 12

\int \frac{4}{u - 2} d u = 8 (u - 2 + 2 ln u - 2)

= \frac{1}{3} (u + 4 u + 8 ln u - 2 - 12 + 8 (u - 2 + 2 ln u - 2))

u = 3 x

代回

= \frac{1}{3} (3 x + 4 3 x + 8 ln 3 x - 2 - 12 + 8 (3 x - 2 + 2 ln 3 x - 2))

化简

\frac{1}{3} (3 x + 4 3 x + 8 ln 3 x - 2 - 12 + 8 (3 x - 2 + 2 ln 3 x - 2)) : \frac{1}{3} (3 x + 123 x + 24 ln 3 x - 2 - 28)

= \frac{1}{3} (3 x + 123 x + 24 ln 3 x - 2 - 28)

解答补常数

= \frac{1}{3} (3 x + 123 x + 24 ln 3 x - 2 - 28) + C

作图

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