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

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

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

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

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

Solution steps

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

Apply Integration By Parts

: ln^{2} (x) - \int \frac{2 ln ( x ) - ln ^{2} ( x )}{x} d x

= ln^{2} (x) - \int \frac{2 ln ( x ) - ln ^{2} ( x )}{x} d x

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

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

Simplify

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

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

Add a constant to the solution

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

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