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$$\int_0^1 \frac{1}{({1-x^6})^{1/6}} dx $$ 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.

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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 $$\int_0^1\frac1{(1-x^n)^{1/n}}\,dx=\frac\pi n\csc\frac\pi n$$

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  • $\begingroup$ @SagunDevkota Please check your work. $\endgroup$ Commented Jan 15, 2021 at 7:25

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