The value of `int(dx)/(x^n(1+x^n)^(1/ n))` is equal to

Options

A. `(1)/(1-n){1+(1)/(x^(n))}^(1-(1)/(n))+C`
B. `(1)/(1+n){1-(1)/(x^(n))}^(1-(1)/(n))+C`
C. `-(1)/(1-n){1-(1)/(x^(n))}^(1-(1)/(n))+C`
D. `-(1)/(1+n){1+(1)/(x^(n))}^(1-(1)/(n))+C`

Answer

A

Explanation

Let `I=int(1)/(x^(n)(1+x^(n))^(1//n))dx=int(1)/(x^(n+1)(1+(1)/(x^(n)))^(1//n))dx`
`rArrI=int(1+x^(-n))^(-1//n)x^(-n-1)dx`
`rArrI=int=-(1)/(n)(1+x^(-n))^(-1//n)d(1+x^(-n))`
`rArr =-(1)/(n)((1+x^(-n))^(-(1)/(n)+1))/((-(1)/(n)+1))+C=(1)/(1-n){1+(1)/(x^(n))}^(1-(1)/(n))+C`