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

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

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

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

2 (- ln x + ln 2 x - 1) + C

Solution steps

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

Apply u-substitution

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

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

Take the partial fraction of

\frac{1}{u ( 2 u - 1 )} : - \frac{1}{u} + \frac{2}{2 u - 1}

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= 2 (- ln ∣ u ∣ + ln ∣ 2 u - 1 ∣)

Substitute back

u = x

= 2 (- ln x + ln 2 x - 1)

Add a constant to the solution

= 2 (- ln x + ln 2 x - 1) + C

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