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

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

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

\int ln (x) \frac{1}{x} d x

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

Solution steps

\int ln (x) \frac{1}{x} d x

Simplify

ln (x) \frac{1}{x} : \frac{ln ( x )}{x}

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

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

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

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

Substitute back

u = ln (x)

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

Add a constant to the solution

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

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