What is the integration of this function? [closed]

Question

$$\int_0^1 \frac{1}{({1-x^6})^{1/6}} dx $$ I have been trying to integrate this function on given limit but I am struggling to find a way to start. How do I integrate this and such type of functions.

Answer 1

Substituting $u=1-x^6$ gives $$I=\frac16\int_0^1u^{-1/6}(1-u)^{-5/6}\,du=\frac16B(1/6,5/6)=\frac\pi{6\sin\pi/6}=\frac\pi3$$ More generally $$\int_0^1\frac1{(1-x^n)^{1/n}}\,dx=\frac\pi n\csc\frac\pi n$$