`int ln(root(3)(x)) dx` Evaluate the integral

`int ln (root(3)(x)) dx`

To evaluate, apply integration by parts  `int udv = uv - int vdu` .

So let

`u = ln (root(3)(x))=ln (x^(1/3))=1/3ln(x)`

and

`dv = dx`

Then, differentiate u and integrate dv.

`du = 1/3*1/x dx= 1/(3x) dx`

and

`v=intdx = x`

And, plug-in them to the formula. So the integral becomes:

`int ln (root(3)(x))dx`

`= 1/3ln (x) *x- int x * 1/(3x)dx`

`= 1/3xln(x) - int 1/3dx`

`=1/3xln(x) - 1/3x+C`

Therefore,  `int ln (root(3)(x)) dx =1/3xln(x)- 1/3x + C` .