What is the integral of (ln^3(x))/2 ?

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

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integral of (ln^3(x))/2

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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