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

The integral of ln^2(2x^2) is x(ln(2)+2ln(x))^2-4(x(ln(2)+2ln(x))-2x)+C

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integral of ln^2(2x^2)

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

x (ln (2) + 2 ln (x))^{2} - 4 (x (ln (2) + 2 ln (x)) - 2 x) + C

Solution steps

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

Apply Integration By Parts

= x ln^{2} (2 x^{2}) - \int 4 ln (2 x^{2}) d x

\int 4 ln (2 x^{2}) d x = 4 (x (ln (2) + 2 ln (x)) - 2 x)

= x ln^{2} (2 x^{2}) - 4 (x (ln (2) + 2 ln (x)) - 2 x)

Simplify

ln (2 x^{2}) : 2 ln (x) + ln (2)

= x (ln (2) + 2 ln (x))^{2} - 4 (x (ln (2) + 2 ln (x)) - 2 x)

Add a constant to the solution

= x (ln (2) + 2 ln (x))^{2} - 4 (x (ln (2) + 2 ln (x)) - 2 x) + C

What is the integral of ln^2(2x^2) ?
The integral of ln^2(2x^2) is x(ln(2)+2ln(x))^2-4(x(ln(2)+2ln(x))-2x)+C