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受欢迎的微积分 >

integral of ((x+1))/(xsqrt(x-2))

解答

\int \frac{( x + 1 )}{x x - 2} d x

解答

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

求解步骤

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

使用换元积分法

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

乘开

\frac{u ^{2} + 3}{u ^{2} + 2} : \frac{u ^{2}}{u ^{2} + 2} + \frac{3}{u ^{2} + 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

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

\int \frac{u ^{2}}{u ^{2} + 2} d u = - 2 arctan (\frac{u}{2}) + u

\int \frac{3}{u ^{2} + 2} d u = \frac{3}{2} arctan (\frac{u}{2})

= 2 (- 2 arctan (\frac{u}{2}) + u + \frac{3}{2} arctan (\frac{u}{2}))

u = x - 2

代回

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

化简

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

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

解答补常数

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

作图

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