Integrate following function w.r.t.x
`(i) int(dx)/(9+x^(2))` `(ii) int(dx)/(sqrt(a^(2)-4x^(2)))`
`(iii) int(sqrt(x)dx)/(sqrt(a^(3)-x^(3)))`

Explanation

`(i) l=int(dx)/(9+x^(2))=int(dx)/(3^(2)+x^(2))=(1)/(3)tan^(-1)((x)/(3))+C`
`(ii) l=int(dx)/(sqrt(a^(2)-4x^(2)))=int(dx)/(2sqrt(((a)/(2))^(2)-x^(2)))=(1)/(2)sin^(-1){(2x)/(a)}+C`
`(iii) l=intsqrt((x)/(a^(3)-x^(3)))dx=int(sqrt(x))/(sqrt((a^((3)/(2)))^(2)-(x^((3)/(2)))^(2)))dx` Put `x^((3)/(2))=t`
`implies(3)/(2)x^((1)/(2))dx=dt`
`:.l=(2)/(3)int(dt)/(sqrt((a)^((3)/(2))-t^(2)))=(2)/(3)sin^(-1){(t)/(a^((3)/(2)))}+C`
`:.l=(2)/(3)sin^(-1){sqrt((x^(3))/(a^(3)))}+C`