`intcos xsqrt(9-sin ^(2)x)+dx=?`

Options

A. `(1)/(2)sin xsqrt(9-sin ^(2)x)+(9)/(2)sin^(-1)""((sinx)/(3))+C`
B. `(sinx)/(2)sqrt(9-sin ^(2)x)+(9)/(2)sin^(-1)""((sinx)/(3))+C`
C. `(1)/(2)cos x sqrt(9-sin ^(2)x)+(9)/(2)sin^(-1)""((sinx)/(3))+C`
D. None of these

Answer

A

Explanation

Put `sin x=t and cos x dx=dt . `then ,
`I=int sqrt(9-t^(2))dt =(t)/(2)sqrt(9-t^(2))+(9)/(2)sin^(-1)""(t)/(3) +C`
`=(sin x)/(2)sqrt(9-sin^(2)x)+(9)/(2) sin ^(-1) ((sin x)/(3))+C.`