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int(x^(2)+1)/(xsqrt(x^(2)+2x-1)sqrt(1-x^...

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

A

`2sin^(-1)sqrt(x-(1)/(x)+2)+c`

B

`2cos^(-1)sqrt(x-(1)/(x)+2)+c`

C

`sin^(-1)sqrt(x-(1)/(x)+2)+c`

D

`cos^(-1)sqrt(x-(1)/(x)+2)+c`

Text Solution

Verified by Experts

The correct Answer is:
A
`I=int(x^(2)+1)/(xsqrt(x^(2)+2x-1)sqrt(1-x^(2)-x))dx`
`=int(1+(1)/(x^(2)))/(sqrt(x+2-(1)/(x))sqrt((1)/(x)-x-1))dx`
`"Let "t=sqrt(x+2-(1)/(x))`
`therefore" "dt=(1)/(2sqrt(x+2-(1)/(x)))(1+(1)/(x^(2)))dx`
`therefore" "I=int(2dt)/(sqrt(1-t^(2)))`
`=2sin^(-1)t+c`
`=2sin^(-1)sqrt(x-(1)/(x)+2)+c`
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