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int(x(x-1))/((x^(2)+1)(x+1)sqrt(x^(3)+x^...

`int(x(x-1))/((x^(2)+1)(x+1)sqrt(x^(3)+x^(2)+x))=(1)/(2)log|(sqrt(x+(1)/(x)+1-1))/(sqrt(x+(1)/(x)+1+1))|-A+c.`
Then the value of A is equal to

A

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

B

`tan^(-1)sqrt(x+(1)/(x)+1)`

C

`cot^(-1)sqrt(x+(1)/(x))`

D

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

Text Solution

Verified by Experts

The correct Answer is:
B
We have `int(x(x-1))/((x^(2)+)(x+1)sqrt(x^(3)+x^(2)+x))dx`
`=int(x(x^(2)-1))/((x^(2)+1)(x+1)^(2)sqrt(x^(3)+x^(2)+x))dx`
`=int(x^(3)(1-(1)/(x^(2))))/(x^(3)(x+(1)/(x))(sqrtx+(1)/(sqrtx))^(2)sqrt(x+(1)/(x)+1))dx`
`=int((1-(1)/(x^(2))))/((x+(1)/(x))(x+(1)/(x)+2)sqrt(x+(1)/(x)+1))dx`
`I=int(2t)/((t^(2)-1)(t^(2)+1)sqrt(t^(2)))dt`
where `x+(1)/(x)+1=t^(2)`
`=int(2)/((t^(2)-1)(t^(2)+1))dt`
`=int(1)/(t^(2)-1)dt-int(1)/(t^(2)+1)dt`
`=(1)/(2)log|(t-1)/(t+1)|-tan^(-1)t+c`
`=(1)/(2)log|(sqrt(x+(1)/(x)+1)-1)/(sqrt(x+(1)/(x)+1+1))|-tan^(-1)sqrt(x+(1)/(x)+1)+c`
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