What is the integral of (ln(ln(5x)))/(5x) ?

The integral of (ln(ln(5x)))/(5x) is 1/5 (ln(5x)ln(ln(5x))-ln(5x))+C

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integral of (ln(ln(5x)))/(5x)

\int \frac{ln ( ln ( 5 x ) )}{5 x} d x

\frac{1}{5} (ln (5 x) ln (ln (5 x)) - ln (5 x)) + C

Solution steps

\int \frac{ln ( ln ( 5 x ) )}{5 x} d x

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{ln ( ln ( 5 x ) )}{x} d x

Apply u-substitution

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

= \frac{1}{5} \cdot \int \frac{ln ( ln ( u ) )}{u} d u

Apply u-substitution

: \int ln (v) d v

= \frac{1}{5} \cdot \int ln (v) d v

Apply Integration By Parts

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

= \frac{1}{5} (v ln (v) - \int 1 d v)

\int 1 d v = v

= \frac{1}{5} (v ln (v) - v)

Substitute back

= \frac{1}{5} (ln (5 x) ln (ln (5 x)) - ln (5 x))

Add a constant to the solution

= \frac{1}{5} (ln (5 x) ln (ln (5 x)) - ln (5 x)) + C

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\frac{\partial}{\partial z} (- x y)

t y^{^{''}} + y^{^{'}} = 0

f (x) = 7 x - 4 x^{2}, (- 3, - 57)

\frac{d}{d x} (e^{x} (2 x^{4} - 1))

What is the integral of (ln(ln(5x)))/(5x) ?
The integral of (ln(ln(5x)))/(5x) is 1/5 (ln(5x)ln(ln(5x))-ln(5x))+C