Evaluate: `int(sqrt(tanx)+sqrt(cotx))dx`

Answer

`sqrt(2)"tan"^(-1)((sqrt("tan x")-sqrt("cot x"))/(sqrt(2)))+c`

Explanation

Let `" "I=int(sqrt(tanx)+sqrt(cotx))dx=int(tanx+1)/(sqrt(tanx))dx` ltBrgt Put `" "tan x = t^(2) rArr sec^(2)x dx = 2t dt`
`rArr dx=(2t)/(1+t^(4))dt`
`therefore" "I=int(t^(2)+1)/(sqrt(t^(2)))*(2t)/(t^(4)+1)dt=2int(t^(2)+1)/(t^(4)+1)dt`
`=2int(1+(1)/(t^(2)))/(t^(2)+(1)/(t^(2))-2+2)dt=2 int(1+(1)/(t^(2)))/((t-(1)/(t))^(2)+(sqrt(2))^(2))dt`
Put `" "t-(1)/(t)=u rArr 1+(1)/(t_(2))dt=du`
`therefore" "I=2int(du)/(u^(2)+(sqrt(2))^(2))`
`rArr" "I=(2)/(sqrt(2))tan^(-1)((u)/(sqrt(2)))+c`
`" "=sqrt(2)tan^(-1)((sqrt(tan x)-sqrt(cot x))/(sqrt(2)))+c`