What is the integral of 3/(x-sqrt(x)) ?

The integral of 3/(x-sqrt(x)) is 6ln|sqrt(x)-1|+C

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integral of 3/(x-sqrt(x))

\int \frac{3}{x - x} d x

6 ln x - 1 + C

Solution steps

\int \frac{3}{x - x} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{x - x} d x

Apply u-substitution

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

= 3 \cdot \int \frac{2}{u - 1} d u

Take the constant out:

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

= 3 \cdot 2 \cdot \int \frac{1}{u - 1} d u

Apply u-substitution

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

= 3 \cdot 2 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 3 \cdot 2 ln ∣ v ∣

Substitute back

= 3 \cdot 2 ln x - 1

Simplify

= 6 ln x - 1

Add a constant to the solution

= 6 ln x - 1 + C

\int x^{2} \cdot \frac{1}{4 x ^{2} + 9} d x

x^{2} + x + 1

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

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

\int - 3 (x + 2)^{- 2} d x

What is the integral of 3/(x-sqrt(x)) ?
The integral of 3/(x-sqrt(x)) is 6ln|sqrt(x)-1|+C