Anti-derivative of [x^(1/2)+(1)/(6)(x^(1/2)]^2

That's sqrt of x, plus 1 over 6 sqrt of x, all to the power of 2.

and what is is usual method to go about finding the anti-derivative of this kind of function?

Thanks.

the expression becomes

x + 1/(36x) + 1/3

int = 1/2 x^2 + 1/36 * ln x + 1/3 x + C