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

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

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

\int \frac{3 ln ( x )}{x} d x

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

Solution steps

\int \frac{3 ln ( 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{ln ( x )}{x} d x

Apply u-substitution

: \int u d u

= 3 \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = ln (x)

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

Simplify

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

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

Add a constant to the solution

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

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