Solving the integral $\int_0^{\frac\pi2}\frac{dx}{1+\sin^2(\tan x)}$

Asked Dec 29 '14 at 08:10

Active Dec 29 '14 at 08:33

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$$\int_0^{\frac\pi2}\frac{dx}{1+\sin^2(\tan x)}$$

First, I tried to set

$$t=\tan x$$

Then I got $$\int_0^\infty\frac{dt}{(1+t^2)(1+\sin^2t)}$$

applied a trig identity, $1+ \sin^2t=\frac{1}{2}(3-\cos2 t)$

I got

$$\int_0^\infty\frac{dt}{(1+t^2)\left(\frac12(3-\cos 2t)\right)}$$

I don't know how to keep going to solve it.

can some one give me a hint?

Thanks

edited Dec 29 '14 at 08:33

asked Dec 29 '14 at 08:10

Simple

1

$1+ \sin^2t=\frac{1}{2}(3-\cos t)$? it should be $1+ \sin^2t=\frac{1}{2}(3-\cos 2t)$ – Krish Dec 29 '14 at 08:30
1

@Krish yes, you are right. But I still don't have any good idea to solve it – Simple Dec 29 '14 at 08:32
Do you want to have a closed form solution to this integral? Because after looking at this Wolfram alpha answer, I am not sure there is one. – Samrat Mukhopadhyay Dec 29 '14 at 08:33
1

@M.N.C.E. thanks for sharing the link – Simple Dec 29 '14 at 08:44

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