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

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

A

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

B

`-2"tan"^(-1)(1)/(sqrt(2e^(x)-1))+c`

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
A, B, D
`I=int(dx)/(sqrt(2e^(x)-1))`
`"Let " 2e^(x)-1=t^(2)`
` :. 2e^(x)dx=2t dt`
` :. I=int(t dt)/((t^(2)+1)/(2)*t)`
`=2tan^(-1)t+C`
`=2tan^(-1)sqrt(2e^(x)-1)+C`
`=-2"tan"^(-1)(1)/(sqrt(2e^(x)-1))+C`
`"Also, " tan^(-1)sqrt(2e^(x)-1)=sec^(-1)sqrt(2e^(x))`
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Knowledge Check

  • The value of int (dx)/(sqrt(e^(2x)-1)) is equal to -

    A
    `sin^(-1)(e^(x))+c`
    B
    `cos^(-1)(e^(x))+c`
    C
    `tan^(-1)(e^(x))+c`
    D
    `sec^(-1)(e^(x))+c`

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