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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

\int 2 ln (x) d x = 2 (x ln (x) - x)

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

Simplify

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

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

Add a constant to the solution

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

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