Evaluate : (i) `intsqrt(9-x^(2))dx`
(ii) `intsqrt(1-4x^(2))dx`
(iii) `intsqrt(16-9x^(2))dx`

Explanation

We know that `intsqrt(a^(2)-x^(2))dx=(x)/(2)sqrt(a^(2)-x^(2))+(a^(2))/(2)"sin"^(-1)(x)/(a)+C`.
`therefore` (i) `intsqrt(9-x^(2))dx=intsqrt(3^(2)-x^(2))dx`
`=(x)/(2)sqrt(9-x^(2))+(9)/(2)"sin"^(-1)(x)/(3)+C`.
(ii) `sqrt(1-4x^(2))dx=2intsqrt(((1)/(4)-x^(2)))dx=2int{sqrt(((1)/(2))^(2)-x^(2))}dx`
`=2[(x)/(2)sqrt((1)/(4)-x^(2))+(1)/(8)sin^(-1)((x)/((1//2)))]+C`
`=(x)/(2)sqrt(1-4x^(2))+(1)/(4)sin^(-1)(2x)+C`.
(iii) `intsqrt(16-9x^(2))dx=3int{sqrt(((16)/(9)-x^(2)))}dx=3int{sqrt(((4)/(3))^(2)-x^(2))}dx`
`=3[(x)/(2)sqrt((16)/(9)-x^(2))+(8)/(9)"sin"^(-1)(x)/((4//3))]+C`
`(x)/(2)sqrt(16-9x^(2))+(8)/(3)sin^(-1)((3x)/(4))+C`.