Evaluate: `int1/(x^(1//2)+x^(1//3))\ dx`

Explanation

`(1)/(x^(1//2)+x^(1//3))=(1)/(x^(1//3)(1+x^(1//6)))`
Let `x=t^(6)impliesdx=6t^(5) dt`
` :. int(1)/(x^(1//2)+x^(1//3))dx=int(1)/(x^(1//3)(1+x^(1//6)))dx`
`=int(6t^(5))/(t^(2)(1+t))dt`
`=6int(t^(3))/((1+t))dt`
On dividing, we obtain
`int(1)/(x^(1//2)+x^(1//3))dx=6int{(t^(2)-t+1)-(1)/(1+t)}dt`
`=6[((t^(3))/(3))-((t^(2))/(3))+t-log|1+t|]+C`
`=2x^(1//2)-3x^(1//3)+6x^(1//6)-6 log(1+x^(1//6))+C`