Find the integral `int (ln x)^2/x dx`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral `int (ln x)^2/x dx` has to be determined.

This can be done by substitution.

Let y = ln x, taking the derivative with respect to x:

`dy/dx = 1/x`

or `dy = dx/x`

Now substitute this in the original integral

`int (ln x)^2/x dx`

= `int y^2 dy`

= `y^3/3`

As y = ln x, the required integral is `(lnx)^3/3`

The integral `int (ln x)^2/x dx = (lnx)^3/3`

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