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

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

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

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

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

Solution steps

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

y^{^{'}} = 6 y^{2} + x y^{2}, y (0) = 1

\frac{d ^{2}}{d x ^{2}} (\frac{2000 t}{4 t + 75})

\int (y cos (x) + 2 x e^{y}) d x

x y = 2

\frac{d}{d x} ((5 x)^{l n (x)})

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