If `int(dx)/(x^2(x^n+1)^((n-1)/n))=-(f(x))^(1/n)+C` then `f(x)` is (A) `1+x^n` (B) `1+x^-n` (C) `x^n+x^-n` (D) `x^n-x^-n`

Options

A. `(1+x^(n))`
B. `1+x^(-n) `
C. `x^(n)+x^(-n) `
D. none of these

Answer

B

Explanation

`"We have " int(dx)/(x^(2)(x^(n)+1)^((n-1)//n))=int(dx)/(x^(2)x^(n-1)(1+(1)/(x^(n)))^((n-1)//n))`
`=int(dx)/(x^(n+1)(1+x^(-n))^((n-1)//n))`
`"Put " 1+x^(-n)=t`
` :. -nx^(-n-1)dx=dt " or " (dx)/(x^(n+1))=-(dt)/(n)`
` :. int(dx)/(x^(2)(x^(n)+1)^((n-1)//n))=-(1)/(n)int(dt)/(t^((n-1)//n)) `
`=-(1)/(n)int t^(((1)/(n)-1))dt `
`=-(1)/(n)(t^((1)/(n)-1+1))/((1)/(n)-1+1)+C`
`=-t^(1//n)+C`
`=-(1+x^(-n))^(1//n)+C`