Evaluate: `int1/(x^(2/3)sqrt(x^(2/3)-4))dx`

Explanation

`∫ frac{1}{x^(2/3) sqrt(x^(2/3)-4)}dx`

Putting, `x^(2/3)-4=t^2`

`=>frac{2}{3}n^(-1/3)dx=2dt`

and `x^(2/3)-4=t^2=>x^(1/3)=sqrt(t^2+4)`

`int(frac{1}{x^(1/3).x^(1/3) sqrt(x^(2/3)-4)})dx`

`=>int (frac{3t dt}{sqrt(t^2+4).sqrt(t^2)})dx`

`=>3 int(frac{1}{sqrt(4+t^2)})dt=>3 ln[t+sqrt(4+t^2)]`

`=>3ln[n^(1/3)+sqrt(x^(2/3)-4)]`