`int1/(sqrt(8+3x-n^2))dx`

Options

A. `(2)/(3)sin^(-1)((2x-1)/(sqrt(41)))+C`
B. `(3)/(2)sin^(-1)((2x-3)/(sqrt(41)))+C`
C. `(1)/(sqrt(41))sin^(-1)((2x-3)/(sqrt(41)))+C`
D. `sin^(-1)((2x-3)/(sqrt(41)))+C`

Answer

D

Explanation

Let `I=int(1)/(sqrt(8+3x-x^(2)))dx`
`=int(1)/(sqrt(8-[x^(2)-3x+((3)/(2))^(2)-((3)/(2))^(2)]))dx`
`=int(1)/(sqrt(8-[(x-(3)/(2))^(2)-(9)/(4)]))dx`
`=int(1)/(sqrt(8+(9)/(4)-(x-(3)/(2))^(2)))dx`
`=int(1)/(sqrt(((sqrt(41))/(2))^(2)-(x-(3)/(2))^(2)))dx`
Let `x-(3)/(2)=t rArr dx=dt`
`therefore" "I=int(1)/(sqrt(((sqrt(41))/2)^(2)-t^(2)))dt=sin^(-1)((t)/((sqrt(41))/(2)))+C`
`=sin^(-1)((2x-3)/(sqrt(41)))+C`