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int(dx)/(sqrt(1-e^(2x))) is equal to...

`int(dx)/(sqrt(1-e^(2x)))` is equal to

A

`log|e^(-x)+sqrt(e^(-2x)-1)|+c`

B

`log|e^(x)+sqrt(e^(2x)-1)|+c`

C

`-log|e^(-x)+sqrte^(-2x)-1)|+c`

D

`-log|e^(-2x)+sqrt(e^(-2x)-1)|+c`

Text Solution

Verified by Experts

The correct Answer is:
C
Let `l=int(dx)/(sqrt(1-e^(2x)))`
`=int(e^(-x))/(sqrt(e^(-2x)-1))dx`
Put `e^(-x)=t`
`impliese^(-x)dx=-dt`
`thereforel=-int(dt)/(sqrt(t^(2)-1))`
`=-log|t+sqrt(t^(2)-1)|+c`
`=-log|e^(-x)+sqrt(e^(-2x)-1)|+c`
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