Evaluate the following integral: $$\int\dfrac1{1+\sqrt{\tan x}}\mathrm dx.$$

I know this question has a solution, but I haven't the slightest idea how to do it.

  • 6
    $\begingroup$ set $t=\sqrt{tan(x)}$ $\endgroup$ – Dr. Sonnhard Graubner Sep 29 '14 at 19:03
  • $\begingroup$ Following Dr. Sonnhard Graubner's method leads to some "work" to find the correct process. Although a good method, in this case I'll take the occasional shortcut and post the answer: "wolframalpha.com/input/?i=\int+\frac{1}{1+%2B+\sqrt{\tan%28x%29}}+dx+" $\endgroup$ – Leucippus Sep 29 '14 at 19:09

$t=\sqrt{\tan(x)}$ then we get $x=\arctan(t^2)$ and $dx=\frac{2tdt}{1+t^4}dt$ and we get the integral $\int\frac{2tdt}{(1+t)(1+t^4)}$


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