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A value of alpha such that int(al...

A value of `alpha` such that `int_(alpha)^(alpha+1) (dx)/((x+alpha)(x+alpha+1))="loge"((9)/(8))` is

A

`-2`

B

`(1)/(2)`

C

`-(1)/(2)`

D

2

Text Solution

Verified by Experts

The correct Answer is:
A
Let I `int_(alpha)^(alpha+1)(dx)/((x+a)(x+alpha1))`
`int_(alpha)^(alpha+1)((x+alpha+1)-(x+alpha))/((x+alpha)(x+alpha+1))`dx
`=int_(alpha)^(alpha+1)((1)/(x+alpha)-(1)/(x+alpha+1)) dx`
`=[log_(e)(x+alpha)-log_(e)(x+alpha+1)]_(alpha)^(alpha+1)`
`=[log_(e)((x+alpha)/(x+alpha+1))]_(alpha)^(alpha+1)`
`="log"_(e)(2alpha+1)/(2alpha+2)-"log"_(e)(2alpha)/(2alpha+1)`
`=log_(e)((2alpha+1)/(2alpha+2)xx(2alpha+1)/(2alpha))="log"_(e)((9)/(8))` (given)
`rArr((2alpha+1)^(2))/(4alpha(alpha+1))=(9)/(8)rArr8 [4alpha+1] = 36 (alpha^(2)+alpha)`
`rArr8alpha^(2)+8alpha+2=9alpha^(2)+9alpha`
`rArralpha^(2)+alpha-2=0`
` rArr(alpha+2)(alpha-1)=0`
`rArralpha=1,-2`
From the options we get `alpha -=2`
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