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

The integral of (sqrt(x))/(2-sqrt(x)) is -x-4sqrt(x)-8ln|-sqrt(x)+2|+12+C

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

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

- x - 4 x - 8 ln - x + 2 + 12 + C

Solution steps

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

Apply u-substitution

: \int - \frac{2 ( u ^{2} - 4 u + 4 )}{u} d u

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

Expand

\frac{u ^{2} - 4 u + 4}{u} : u - 4 + \frac{4}{u}

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

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

\int 4 d u = 4 u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

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

Substitute back

u = 2 - x

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

Simplify

- 2 (\frac{( 2 - x ) ^{2}}{2} - 4 (2 - x) + 4 ln 2 - x) : - x - 4 x - 8 ln - x + 2 + 12

= - x - 4 x - 8 ln - x + 2 + 12

Add a constant to the solution

= - x - 4 x - 8 ln - x + 2 + 12 + C

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What is the integral of (sqrt(x))/(2-sqrt(x)) ?
The integral of (sqrt(x))/(2-sqrt(x)) is -x-4sqrt(x)-8ln|-sqrt(x)+2|+12+C