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The value of the integral int(dx)/(x^(n)...

The value of the integral `int(dx)/(x^(n)(1+x^(n))^(1//n)), n in N` is

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`

Text Solution

Verified by Experts

The correct Answer is:
A
`int(dx)/(x^(n)(1+x^(n))^(1//n))=int(dx)/(x^(n).x((1)/(x^(n))+1)^(1//n))`
`=int(dx)/(x^(n+1)((1)/(x^(n))+1)^(1//n))`
`"Put "(1)/(x^(n))+1=t`
`rArr-(n)/(x^(n+1))dx=dt`
`rArr" "-(1)/(n)int(dt)/(t^(1//n))=-(1)/(n)int t^(-1//n)dt=-(1)/(n).(t^(1-(1)/(n)))/((1-(1)/(n)))+C`
`=(1)/((1-n))(1+(1)/(x^(n)))^(1-(1)/(n))+C`
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