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

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

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

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

\frac{1}{2} ln^{2} (x) + \frac{1}{3} ln^{3} (x) + C

Solution steps

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

Apply Integration By Parts

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

= \frac{1}{2} ln^{2} (x) (1 + ln (x)) - \int \frac{ln ^{2} ( x )}{2 x} d x

\int \frac{ln ^{2} ( x )}{2 x} d x = \frac{1}{6} ln^{3} (x)

= \frac{1}{2} ln^{2} (x) (1 + ln (x)) - \frac{1}{6} ln^{3} (x)

Simplify

\frac{1}{2} ln^{2} (x) (1 + ln (x)) - \frac{1}{6} ln^{3} (x) : \frac{1}{2} ln^{2} (x) + \frac{1}{3} ln^{3} (x)

= \frac{1}{2} ln^{2} (x) + \frac{1}{3} ln^{3} (x)

Add a constant to the solution

= \frac{1}{2} ln^{2} (x) + \frac{1}{3} ln^{3} (x) + C

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