The value of ∫ 1/(x^n≤ft(1+x^n)^1 / n) d x,(n ∈ N), is

Let

$I =\int \frac{1}{x^n\left(1+x^n\right)^{1 / n}} d x=\int \frac{1}{x^{n+1}\left(1+\frac{1}{x^n}\right)^{1 / n}} d x$

$\Rightarrow I =\int\left(1+x^{-n}\right)^{-1 / n} x^{-n-1} d x$

$\Rightarrow I =-\frac{1}{n} \int\left(1+x^{-n}\right)^{-1 / n} d\left(1+x^{-n}\right)$

$\Rightarrow I=-\frac{1}{n} \frac{\left(1+x^{-n}\right)^{-\frac{1}{n}+1}}{\left(-\frac{1}{n}+1\right)}+C=\frac{1}{1-n}\left\{1+\frac{1}{x^n}\right\}^{1-\frac{1}{n}}+C$