Problem: $$\int_{0}^{\frac{\pi}{2}} \frac{(\sec x)^{\frac{1}{3}}}{(\sec x)^{\frac{1}{3}}+(\tan x)^{\frac{1}{3}}} dx$$
My attempt:
I tried applying the property: $\int_{0}^{a} f(x)dx$ = $\int_{0}^{a} f(a-x)dx$ but got nowhere since the denominator changes. Even on adding the two integrals by taking LCM of the denominators, the final expression got more complicated because the numerator and denominator did not have any common factor.
I also tried dividing numerator and denominator by $(secx)^{\frac{1}{3}}$ to get $$\int_{0}^{\frac{\pi}{2}} \frac{1}{1+(\sin x)^{\frac{1}{3}}} dx$$ and then tried substituting $sinx$ = $t^3$ to get a complicated integral in $t$, which I couldn't evaluate.
How do you evaluate this integral? (PS: If possible, please evaluate this without using special functions since this is a practice question for an entrance exam and we've only learnt some basic special functions and the gamma function.)