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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{12} (- 18 x + 122 x + 2 (2 x + 1)^{3} - 18 ln 2 x + 1 + 21) + C

求解步骤

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

使用换元积分法

= \int \frac{u + 2}{4 ( u + 1 )} d u

提出常数:

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

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

乘开

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

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

\int \frac{u}{u + 1} d u = - 3 u + \frac{2}{3} (u + 1)^{3} - 2 ln u + 1 + 3

\int \frac{2}{u + 1} d u = 4 (u + 1 - ln u + 1)

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

u = 2 x

代回

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

化简

\frac{1}{4} (- 3 \cdot 2 x + \frac{2}{3} (2 x + 1)^{3} - 2 ln 2 x + 1 + 3 + 4 (2 x + 1 - ln 2 x + 1)) : \frac{1}{12} (- 18 x + 122 x + 2 (2 x + 1)^{3} - 18 ln 2 x + 1 + 21)

= \frac{1}{12} (- 18 x + 122 x + 2 (2 x + 1)^{3} - 18 ln 2 x + 1 + 21)

解答补常数

= \frac{1}{12} (- 18 x + 122 x + 2 (2 x + 1)^{3} - 18 ln 2 x + 1 + 21) + C

作图

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