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

The integral of (ln(ln(5x)))/(xln(5x)) is (ln^2(ln(5x)))/2+C

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

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

\frac{ln ^{2} ( ln ( 5 x ) )}{2} + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

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

= \int \frac{ln ( v )}{v} d v

Apply u-substitution

: \int w d w

= \int w d w

Apply the Power Rule

: \frac{w ^{2}}{2}

= \frac{w ^{2}}{2}

Substitute back

= \frac{ln ^{2} ( ln ( 5 x ) )}{2}

Add a constant to the solution

= \frac{ln ^{2} ( ln ( 5 x ) )}{2} + C

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What is the integral of (ln(ln(5x)))/(xln(5x)) ?
The integral of (ln(ln(5x)))/(xln(5x)) is (ln^2(ln(5x)))/2+C