What is the integral of 1/(x^{1/2)-1} ?

The integral of 1/(x^{1/2)-1} is 2(x^{1/2}-1+ln|x^{1/2}-1|)+C

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integral of 1/(x^{1/2)-1}

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

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

Solution steps

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

Apply u-substitution

: \int \frac{2 u}{u - 1} d u

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

Apply u-substitution

: \int \frac{v + 1}{v} d v

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

Expand

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

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

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= 2 (v + ln ∣ v ∣)

Substitute back

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

Add a constant to the solution

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

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