You need to use integration by parts, hence you should remember the formula such that:

`int udv = uv - int vdu`

`dv= x^(-2/3) =gtv = x^(-2/3+ 1)/(-2/3+1)=gtv = 3x^(1/3)`

`u= ln 5x =gt du= (dx)/x`

`int x^(-2/3)*ln 5x dx =root(3)x*ln 5x - 3int x^(1/3)/x dx`

`int x^(-2/3)* ln 5x dx = root(3)x* ln 5x - 3int x^(1/3-1) dx`

`int x^(-2/3)* ln 5x dx = root(3)x* ln 5x - 3int x^(-2/3) dx`

`int x^(-2/3)* ln 5x dx = root(3)x* ln 5x - 9 root(3) x + c`

**Hence, evaluating the integral using parts yields `int x^(-2/3)* ln 5x dx = root(3)x* ln 5x - 9 root(3) x + c.` **

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