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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{3} (2 x - 1)^{\frac{3}{2}} + 2 x - 1 + C

求解步骤

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

提出常数:

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

= 2 \cdot \int \frac{x}{2 x - 1} d x

使用换元积分法

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

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int u^{2} + 1 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

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

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

\int 1 d u = u

= 2 \cdot \frac{1}{2} (\frac{u ^{3}}{3} + u)

u = 2 x - 1

代回

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

化简

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

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

解答补常数

= \frac{1}{3} (2 x - 1)^{\frac{3}{2}} + 2 x - 1 + C

作图

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