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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int \frac{v + 2}{v} d v

= 2 \cdot \int \frac{v + 2}{v} d v

Expand

\frac{v + 2}{v} : 1 + \frac{2}{v}

= 2 \cdot \int 1 + \frac{2}{v} d v

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 1 d v + \int \frac{2}{v} d v)

\int 1 d v = v

\int \frac{2}{v} d v = 2 ln ∣ v ∣

= 2 (v + 2 ln ∣ v ∣)

Substitute back

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

Add a constant to the solution

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

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