What is the integral of 1/(xsqrt(log_{e)(\sqrt{x))}} ?

The integral of 1/(xsqrt(log_{e)(\sqrt{x))}} is 2sqrt(2)sqrt(ln(x))+C

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integral of 1/(xsqrt(log_{e)(\sqrt{x))}}

\int \frac{1}{x lo g _{e} ( x )} d x

22 ln (x) + C

Solution steps

\int \frac{1}{x lo g _{e} ( x )} d x

Simplify

\frac{1}{x lo g _{e} ( x )} : \frac{2}{x ln ( x )}

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

Take the constant out:

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

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

Apply u-substitution

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

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

Apply the Power Rule

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

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

Substitute back

u = ln (x)

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

Apply radical rule:

= 22 ln (x)

Add a constant to the solution

= 22 ln (x) + C

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What is the integral of 1/(xsqrt(log_{e)(\sqrt{x))}} ?
The integral of 1/(xsqrt(log_{e)(\sqrt{x))}} is 2sqrt(2)sqrt(ln(x))+C