What is the integral of (e^{sqrt(ln(x))})/(2xsqrt(ln(x))) ?

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

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integral of (e^{sqrt(ln(x))})/(2xsqrt(ln(x)))

\int \frac{e ^{l n (x)}}{2 x ln ( x )} d x

e^{2} ln (x) + C

Solution steps

\int \frac{e ^{l n (x)}}{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{e ^{l n (x)}}{x ln ( x )} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{e ^{l n (x)}}{x ln ( x )} = (e) ln^{- \frac{1}{2}} (x) \frac{e}{x}

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

Take the constant out:

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

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

Simplify

ee : (e)^{2}

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

Simplify

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

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

Apply u-substitution

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

= \frac{1}{2} (e)^{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= \frac{1}{2} (e)^{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = ln (x)

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

Simplify

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

= e^{2} ln (x)

Add a constant to the solution

= e^{2} ln (x) + C

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