Question

Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`

Answer 1

Let I = `int sqrt(tanx)/(sinx*cosx)*dx`

Dividing numerator and denominator by cos²x, we get

I = `int (((sqrt(tanx))/(cos^2)))/(((sinx)/(cosx)))*dx`

= `int (sqrt(tanx)*sec^2x)/tanx*dx`

= `int (sec^2x)/sqrt(tanx)*dx`

Put tan x = t
∴ sec²x·dx = dt

∴ I = `int (1)/sqrt(t)*dt`

= `int t^(-1/2)*dt`

= `t^(1/2)/(1/2) + c`

= `2sqrt(t) + c`

= `2sqrt(tanx) + c`.