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

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

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integral of ln^2(x/3)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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