What is the integral of (2+sqrt(x))/(1-sqrt(x)) ?

The integral of (2+sqrt(x))/(1-sqrt(x)) is -x-6sqrt(x)-6ln|-sqrt(x)+1|+7+C

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

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

- x - 6 x - 6 ln - x + 1 + 7 + C

Solution steps

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

Expand

\frac{2 + x}{1 - x} : \frac{2}{1 - x} + \frac{x}{1 - x}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

= - 4 (- 1 + x + ln 1 - x) - x - 2 x - 2 ln - x + 1 + 3

Simplify

- 4 (- 1 + x + ln 1 - x) - x - 2 x - 2 ln - x + 1 + 3 : - x - 6 x - 6 ln - x + 1 + 7

= - x - 6 x - 6 ln - x + 1 + 7

Add a constant to the solution

= - x - 6 x - 6 ln - x + 1 + 7 + C

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What is the integral of (2+sqrt(x))/(1-sqrt(x)) ?
The integral of (2+sqrt(x))/(1-sqrt(x)) is -x-6sqrt(x)-6ln|-sqrt(x)+1|+7+C