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

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

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

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

2 (- 2 ln 2 - x - x) + C

Solution steps

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

Apply u-substitution

: \int \frac{2 u}{2 - u} d u

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

\frac{u}{2 - u} = \frac{2}{2 - u} - 1

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

\int \frac{2}{2 - u} d u = - 2 ln ∣ 2 - u ∣

\int 1 d u = u

= 2 (- 2 ln ∣ 2 - u ∣ - u)

Substitute back

u = x

= 2 (- 2 ln 2 - x - x)

Add a constant to the solution

= 2 (- 2 ln 2 - x - x) + C

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