What is the integral of (ln(x))/x-((ln(x))^2)/x ?

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

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integral of (ln(x))/x-((ln(x))^2)/x

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

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

Solution steps

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

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

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

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

Add a constant to the solution

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

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