What is the integral of (sqrt(2ln(x)-6))/x ?

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

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integral of (sqrt(2ln(x)-6))/x

\int \frac{2 ln ( x ) - 6}{x} d x

\frac{1}{3} (2 ln (x) - 6)^{\frac{3}{2}} + C

Solution steps

\int \frac{2 ln ( x ) - 6}{x} d x

Apply u-substitution

: \int 2 u - 6 d u

= \int 2 u - 6 d u

Apply u-substitution

: \int \frac{v}{2} d v

= \int \frac{v}{2} d v

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

= \frac{1}{2} \cdot \frac{2}{3} (2 ln (x) - 6)^{\frac{3}{2}}

Simplify

\frac{1}{2} \cdot \frac{2}{3} (2 ln (x) - 6)^{\frac{3}{2}} : \frac{1}{3} (2 ln (x) - 6)^{\frac{3}{2}}

= \frac{1}{3} (2 ln (x) - 6)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3} (2 ln (x) - 6)^{\frac{3}{2}} + C

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