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Evaluate: inte^x/(sqrt(1-e^(2x)))dx...

Evaluate: `inte^x/(sqrt(1-e^(2x)))dx`

Text Solution

Verified by Experts

The correct Answer is:
`-log|e^(-x)+sqrt(e^(-2x)-1)|+C`
`I=int(1)/(sqrt(1-e^(2x)))dx=int(1)/(sqrt(1-(1)/(e^(-2x))))dx=int(e^(-x))/(sqrt(e^(-2x)-1))dx`
`=int(e^(-x))/(sqrt((e^(-x))^(2)-1^(2)))dx`
`"Let " e^(-x)=t " or " -e^(-x)dx=dt`
` :. I= -int(dt)/(sqrt(t^(2)-1^(2)))= -log|t+sqrt(t^(2)-1)|+C`
`= -log|e^(-x)+sqrt(e^(-2x)-1)|+C`
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