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

The integral of ln^3(x^2) is 8(xln^3(x)-3(xln^2(x)-2(xln(x)-x)))+C

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integral of ln^3(x^2)

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

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

Solution steps

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

Simplify

ln^{3} (x^{2}) : 8 ln^{3} (x)

= \int 8 ln^{3} (x) d x

Take the constant out:

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

= 8 \cdot \int ln^{3} (x) d x

Apply Integration By Parts

= 8 (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))

= 8 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)))

Add a constant to the solution

= 8 (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 8(xln^3(x)-3(xln^2(x)-2(xln(x)-x)))+C