What is the integral of (ln(sqrt(3x)))/x ?

The integral of (ln(sqrt(3x)))/x is 1/4 ln^2(3x)+C

Popular >

integral of (ln(sqrt(3x)))/x

\int \frac{ln ( 3 x )}{x} d x

\frac{1}{4} ln^{2} (3 x) + C

Solution steps

\int \frac{ln ( 3 x )}{x} d x

Apply u-substitution

= \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

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

Apply the Power Rule

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

Substitute back

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

Simplify

\frac{1}{2} \cdot \frac{ln ^{2} ( 3 x )}{2} : \frac{1}{4} ln^{2} (3 x)

= \frac{1}{4} ln^{2} (3 x)

Add a constant to the solution

= \frac{1}{4} ln^{2} (3 x) + C

- \frac{18}{t ^{3}} + \frac{3}{t ^{2}}

\int x x + 2 d x

\int (x - 1) ln (x) d x

\frac{\partial}{\partial x} (- f (r, x) r sin (x) \cdot sin (x))

f (x) = (- 2 x^{2} - 2)^{3}

What is the integral of (ln(sqrt(3x)))/x ?
The integral of (ln(sqrt(3x)))/x is 1/4 ln^2(3x)+C