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