What is the integral of ((ln(x)-1)ln(x))/x ?

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

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integral of ((ln(x)-1)ln(x))/x

\int \frac{( ln ( x ) - 1 ) ln ( x )}{x} d x

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

Solution steps

\int \frac{( ln ( x ) - 1 ) ln ( x )}{x} d x

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Add a constant to the solution

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

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