What is the integral of (sqrt(ln(4x)))/x ?

The integral of (sqrt(ln(4x)))/x is 2/3 ln^{3/2}(4x)+C

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integral of (sqrt(ln(4x)))/x

\int \frac{ln ( 4 x )}{x} d x

\frac{2}{3} ln^{\frac{3}{2}} (4 x) + C

Solution steps

\int \frac{ln ( 4 x )}{x} d x

Apply u-substitution

= \int 2 u^{2} d u

Take the constant out:

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

= 2 \cdot \int u^{2} d u

Apply the Power Rule

= 2 \cdot \frac{u ^{3}}{3}

Substitute back

u = ln (4 x)

= 2 \cdot \frac{( ln ( 4 x ) ) ^{3}}{3}

Simplify

2 \cdot \frac{( ln ( 4 x ) ) ^{3}}{3} : \frac{2}{3} ln^{\frac{3}{2}} (4 x)

= \frac{2}{3} ln^{\frac{3}{2}} (4 x)

Add a constant to the solution

= \frac{2}{3} ln^{\frac{3}{2}} (4 x) + C

What is the integral of (sqrt(ln(4x)))/x ?
The integral of (sqrt(ln(4x)))/x is 2/3 ln^{3/2}(4x)+C