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intsqrt(e^x-1)dx is equal to (a) 2[sqr...

`intsqrt(e^x-1)dx` is equal to
(a) `2[sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)]+c`
(b) `sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)+c`
(c) `sqrt(e^x-1)+tan^(-1)sqrt(e^x-1)+c`
(d) `2[sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)]+c`

A

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

B

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

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
A
`I=int sqrt(e^(x)-1)dx`
Let ` e^(x)-1=t^(2)" or " e^(x)dx=2t dt " or " dx=(2t)/(t^(2)+1)dt`
` :. I=int t(2t)/(t^(2)+1)dt=int (2t^(2))/(t^(2)+1)dt`
`=int(2(t^(2)+1)-2)/(t^(2)+1)dt=int2dt-int(2t)/(t^(2)+1)`
`=2t-2tan^(-1)t+C`
`=2sqrt(e^(x)-1)-2tan^(-1) sqrt(e^(x)-1)+C`
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