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

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

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

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

2 ln x - 1 + C

Solution steps

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

Apply u-substitution

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

= \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

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 ln ∣ v ∣

Substitute back

= 2 ln x - 1

Add a constant to the solution

= 2 ln x - 1 + C

\frac{d y}{d x} = y^{\frac{2}{3}}, y (1) = 8

\int \frac{1}{41 y ^{2} + 37 y} d y

t^{3} \frac{d y}{d t} + 3 t^{2} y = 6 cos (t), y (π) = 0

2 \frac{d y}{d t} - y = 4 sin (3 t)

f (x) = x^{2} + 7, (1, 8)

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