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

The integral of (sqrt(ln(2x)))/(2x) is 1/3 ln^{3/2}(2x)+C

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integral of (sqrt(ln(2x)))/(2x)

\int \frac{ln ( 2 x )}{2 x} d x

\frac{1}{3} ln^{\frac{3}{2}} (2 x) + C

Solution steps

\int \frac{ln ( 2 x )}{2 x} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{ln ( 2 x )}{x} d x

Apply u-substitution

: \int \frac{ln ( u )}{u} d u

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

Apply u-substitution

: \int v d v

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

Apply the Power Rule

: \frac{2}{3} v^{\frac{3}{2}}

= \frac{1}{2} \cdot \frac{2}{3} v^{\frac{3}{2}}

Substitute back

= \frac{1}{2} \cdot \frac{2}{3} ln^{\frac{3}{2}} (2 x)

Simplify

\frac{1}{2} \cdot \frac{2}{3} ln^{\frac{3}{2}} (2 x) : \frac{1}{3} ln^{\frac{3}{2}} (2 x)

= \frac{1}{3} ln^{\frac{3}{2}} (2 x)

Add a constant to the solution

= \frac{1}{3} ln^{\frac{3}{2}} (2 x) + C

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What is the integral of (sqrt(ln(2x)))/(2x) ?
The integral of (sqrt(ln(2x)))/(2x) is 1/3 ln^{3/2}(2x)+C