Evaluate : (i) `int(dx)/((1+3sin^(2)x))`
(ii) `int(dx)/((3+2cos^(2)x))`

Explanation

(i) Divinding the numerator and denominator by `cos^(2)` x , we get
`int(dx)/((1+3sin^(2)x))=int(sec^(2)x)/(sec^(2)x+3tan^(2)x)dx=int(sec^(2)x)/((1+4tan^(2)x))dx`
`=int(dt)/((1+4t^(2)))` [ putting tan x =t]
`=(1)/(4)int(dt)/([t^(2)+((1)/(2))^(2)])=(1)/(4)*(1)/((1//2))"tan"^(-1)(t)/((1//2))+C`
`=(1)/(2)tan^(-1)(2t)+C=(1)/(2)tan^(1)(2tanx)+C`.
(ii) Dividing the numerator and denominator by `cos^(2)x` , we get
`int(dx)/((3+2cos^(2)))=int(sec^(2)x)/((3sec^(2)x+2))dx=int(sec^(2)x)/(5+3tan^(2)x)dx`
`=int(dt)/((5+3t^(2)))` [ Putting tan x =t]
`=(1)/(3)int(dt)/([t^(2)+((5)/(3))])=(1)/(3)*int(dt)/([t^(2)+(sqrt(5)/(sqrt(3)))^(2)])`
`=(1)/(3)*(1)/((sqrt(5)/(sqrt(3))))"tan"^(-1)(t)/((sqrt(5)/(sqrt(3))))+C=(1)/(sqrt(15))tan^(-1)(sqrt(3t)/(sqrt(5)))+C`
`=(1)/(sqrt(15))tan^(-1)((sqrt(3)tanx)/(sqrt(5)))+C`.