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int(dx^(3))/(x^(3)(x^(n)+1)) equals...

`int(dx^(3))/(x^(3)(x^(n)+1))` equals

A

`(3)/(n)ln((x^(n))/(x^(n)+1))`

B

`(1)/(n)ln((x^(n))/(x^(n)+1))`

C

`(3)/(n)ln((x^(n)+1)/(x^(n)))`

D

`3nln((x^(n+1))/(x^(n)))`

Text Solution

Verified by Experts

The correct Answer is:
A
`int(dx^(3))/(x^(3)(x^(n)+1))`
`=3int(x^(2)dx)/(x^(3)(x^(n)+1))`
`=3int(dx)/(x(x^(n)+1))`
`=int(x^(n-1))/(x^(n)(x^(n)+1))dx`
`=(3)/(n)int(dt)/(t(t+1))`
`=(3)/(n)int((t+1)-t)/(t(t+1))dt`
`=(3)/(n)int((1)/(t)-(1)/(t+1))dt`
`=(3)/(n)(ln t-ln(t+1))+c`
`=(3)/(n)ln((t)/(t+1))+C`
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