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

The integral of 3/(x(ln(x))^{1/2)} is 6ln^{1/2}(x)+C

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integral of 3/(x(ln(x))^{1/2)}

\int \frac{3}{x ( ln ( x ) ) ^{\frac{1}{2}}} d x

6 ln^{\frac{1}{2}} (x) + C

Solution steps

\int \frac{3}{x ln ^{\frac{1}{2}} ( 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{1}{x ln ^{\frac{1}{2}} ( x )} d x

Apply u-substitution

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

= 3 \cdot \int \frac{1}{u ^{\frac{1}{2}}} d u

Apply the Power Rule

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

= 3 \cdot 2 u^{\frac{1}{2}}

Substitute back

u = ln (x)

= 3 \cdot 2 ln^{\frac{1}{2}} (x)

Simplify

= 6 ln^{\frac{1}{2}} (x)

Add a constant to the solution

= 6 ln^{\frac{1}{2}} (x) + C

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What is the integral of 3/(x(ln(x))^{1/2)} ?
The integral of 3/(x(ln(x))^{1/2)} is 6ln^{1/2}(x)+C