What is the integral of 1/(2xsqrt(ln(x))) ?

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

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integral of 1/(2xsqrt(ln(x)))

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

ln (x) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u} d u

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

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = ln (x)

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

Simplify

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

= ln (x)

Add a constant to the solution

= ln (x) + C

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