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

The integral of (ln(ln(3x)))/(3x) is 1/3 (ln(3x)ln(ln(3x))-ln(3x))+C

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

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

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

Solution steps

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

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{ln ( ln ( 3 x ) )}{x} d x

Apply u-substitution

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

= \frac{1}{3} \cdot \int \frac{ln ( ln ( u ) )}{u} d u

Apply u-substitution

: \int ln (v) d v

= \frac{1}{3} \cdot \int ln (v) d v

Apply Integration By Parts

: v ln (v) - \int 1 d v

= \frac{1}{3} (v ln (v) - \int 1 d v)

\int 1 d v = v

= \frac{1}{3} (v ln (v) - v)

Substitute back

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

Add a constant to the solution

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

x \to 3 lim (5 x^{3} - 3 x + x - 6)

\frac{\partial}{\partial x} (am x^{b} m y^{1 - b})

\frac{d}{d x} (arccos (x^{2}))

\int_{0}^{1} \frac{2 x}{e ^{2 x}} d x

y = 2 x^{3} + 3 x^{2} - 12 x + 7

What is the integral of (ln(ln(3x)))/(3x) ?
The integral of (ln(ln(3x)))/(3x) is 1/3 (ln(3x)ln(ln(3x))-ln(3x))+C