How to find the integral $\int_0^1 \frac{x^{1/2}}{1+x^{1/3}}dx$? How to find $$\int_0^1 \frac{x^{1/2}}{1+x^{1/3}}dx$$ ?
My attempt: I made $t^6=x$ and got $\displaystyle \int_0^1 \frac{6t^8}{1+t^2}dt$ and got stuck.

Try writing the numerator as $$6t^8 + 6t^6 - 6t^6 - 6t^4 + 6t^4 + 6t^2 - 6t^2 - 6 + 6.$$
