Question

integration of square root of tanx plus square root of cotx dx

Answer 1

$f(x)=\sqrt{tanx}+\sqrt{cotx}=\sqrt2* \frac{sinx+cosx}{\sqrt{sin2x}}\\\\let\ sin2x=t^2,\ \ cos2x\ dx=(cosx-sinx)(cosx+sinx)dx=t\ dt\\\\\int {f(x)} \, dx=\sqrt2 \int {\frac{dt}{cosx-sinx}} \, dt=\sqrt2 \int {\frac{1}{\sqrt{1-sin2x}}} \, dt\\\\= \sqrt2 \int {\frac{1}{\sqrt{1-t^2}}} \, dt=\sqrt2 Sin^{-1}(\sqrt{sin2x})+K$