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int(1)/((x-1)sqrt(x^(2)-1))dx equals...

`int(1)/((x-1)sqrt(x^(2)-1))dx` equals

A

`-sqrt((x-1)/(x+1))+C`

B

`sqrt((x-1)/(x+1))+C`

C

`sqrt((x+1)/(x-1))+C`

D

`-sqrt((x+1)/(x-1))+C`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `I=int(1)/((x-1)sqrt(x^(2))-1)dx=int(1)/((x-1)^(3//2)sqrt(x+1))dx`
`rArrI=int((x+1)/(x-1))^(3//2)(1)/((x+1)^(2))dx=(1)/(2)int((x-1)/(x+1))^(-3//2)d((x-1)/(x+1))`
`rArrI=-((x-1)/(x+1))^(-1//2)+C=-sqrt((x+1)/(x-1))+C`
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