Evaluate : (i) `int(sinx)/(sqrt(4cos^(2)x-1))dx`
(ii) `int(sec^(2)x)/(sqrt(tan^(2)x-4))dx`

Explanation

(i) Putting `cosx=t and - sin x dx = dt` , we get
`int(sinx)/(sqrt(4 cos^(2)x-1))dx=int(-dt)/(sqrt(4t^(2)-1))=-(1)/(2)int(dt)/(sqrt(t^(2)-(1//2))^(2))`
`=-(1)/(2)*log|t+sqrt(t^(2)-(1)/(4))|+C`
`=-(1)/(2)*log|2t+sqrt(4t^(2)-1)|+C`
`=-(1)/(2)log|2cosx+sqrt(4cos^(2)x-1)|+C`.
Putting `tanx=t and sec^(2)x dx = dt` , we get
`int(sec^(2)x)/(sqrt(tan^(2)x-4))dx=int(dt)/(sqrt(t^(2)-4))=log|t+sqrt(t^(2)-4)|+C`
`=log|tanx+sqrt(tan^(2)x-4)|+C`.