What is the integral of (ln(sqrt(9x)))/x ?

The integral of (ln(sqrt(9x)))/x is 1/4 ln^2(9x)+C

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integral of (ln(sqrt(9x)))/x

\int \frac{ln ( 9 x )}{x} d x

\frac{1}{4} ln^{2} (9 x) + C

Solution steps

\int \frac{ln ( 9 x )}{x} d x

Apply u-substitution

: \int \frac{ln ( u )}{2 u} d u

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

Take the constant out:

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

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

Apply u-substitution

: \int v d v

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

Apply the Power Rule

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

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

Substitute back

= \frac{1}{2} \cdot \frac{ln ^{2} ( 9 x )}{2}

Simplify

\frac{1}{2} \cdot \frac{ln ^{2} ( 9 x )}{2} : \frac{1}{4} ln^{2} (9 x)

= \frac{1}{4} ln^{2} (9 x)

Add a constant to the solution

= \frac{1}{4} ln^{2} (9 x) + C

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

- ln (x)

\frac{d}{d x} ((x - 2) x^{3})

f (x) = 14 - x^{2}, (2, 10)

f (x) = x^{3} + 9 x - 1

What is the integral of (ln(sqrt(9x)))/x ?
The integral of (ln(sqrt(9x)))/x is 1/4 ln^2(9x)+C