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

The integral of (sqrt(x))/(1-sqrt(x)) is -x-2sqrt(x)-2ln|-sqrt(x)+1|+3+C

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

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

- x - 2 x - 2 ln - x + 1 + 3 + C

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Expand

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

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

Apply the Sum Rule:

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

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

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

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= - 2 (\frac{u ^{2}}{2} - 2 u + ln ∣ u ∣)

Substitute back

u = 1 - x

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

Simplify

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

= - x - 2 x - 2 ln - x + 1 + 3

Add a constant to the solution

= - x - 2 x - 2 ln - x + 1 + 3 + C

What is the integral of (sqrt(x))/(1-sqrt(x)) ?
The integral of (sqrt(x))/(1-sqrt(x)) is -x-2sqrt(x)-2ln|-sqrt(x)+1|+3+C