Integral of sqrt tan(x)

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cosec(x)

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Integral of sqrt tan(x)

« on: February 28, 2008, 08:59:07 pm »

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I came across this question last year in MTH620-120 at Melbourne doing UMEP and have never been able to get to work out how to answer it. It is very challenging and kudos to anyone who can show a worked example:

Evaluate
⌠
⌡√tan(x) dx

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Neobeo

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Re: Integral of sqrt tan(x)

« Reply #1 on: February 28, 2008, 09:37:38 pm »

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$\text{Let }u=\sqrt{\tan x}$

$u^2=\tan x$

$2udu=\sec^2 xdx$

$\sec^2 x = \tan^2 x + 1 = u^4+1$

$\therefore \int\sqrt{\tan x} dx = \int\frac{2u^2}{u^4+1} du$

... and I got too lazy to complete the rest of it.

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Ahmad

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Re: Integral of sqrt tan(x)

« Reply #2 on: February 28, 2008, 10:04:30 pm »

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Fair few threads about this on AoPS.

One of them:
http://www.artofproblemsolving.com/Forum/viewtopic.php?t=168301

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