If `intsqrt((x-5)/(x-7))dx = A sqrt(x^(2) - 12 x + 35) + log|x-6 +sqrt(x^(2)-12x+35)| + C`, then

Options

A. -1
B. `1/2`
C. `-1/2`
D. `1`

Answer

D

Explanation

Let `I = intsqrt((x-5)/(x-7))dx`
`=intsqrt(((x-5)(x-5))/((x-7)(x-5))) dx` [rationlising]
`= int(x-5)/(sqrt(x^(2)- 5x-7x +35))dx`
`=int(x-5)/(sqrt(x^(2)-12x+35))dx`
`=1/2int(2x-10)/(sqrt(x^(2)- 12x+35))dx`
`= 1/2int(2x-12)/(sqrt(x^(2)-12x+35))dx+int(1)/(sqrt(x^(2) -12x+35))dx`
`=sqrt(x^(2)-12x+35)+int(1)/(sqrt((x^(2)-12x+36-1)))dx+C`
[ `:'` Let`x^(2)-12x+35=trArr(2x-12)dx=dt`]
`= sqrt(x^(2)-12x+35)+int(1)/(sqrt((x-6)^(2)-1^(2)))dx+C`
`I = sqrt(x^(2)-12x+35)+log|x-6+sqrt(x^(2)-12x+35)|+C`
`:. I = Asqrt(x^(2)-12x+35)+log|x-6+sqrt(x^(2)-12x+35)+C`
On comparing both sides , we get `A = 1`