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Question:

$$\int_0^{\pi/2} \frac {\sqrt{\sin x}}{\sqrt {\sin x} + \sqrt {\cos x}}\mathrm dx.$$

What we did:

we tried using $t=\tan (\frac x2)$ and also dividing both numerator and denominator by $\sqrt {\cos x}$, eventually using the second method we got to this: $\displaystyle \int \frac {2t+2}{t^2+2t-1}-\frac {2}{t^2+2t-1} +\frac {\sqrt{2t(1-t^2)}}{t^2+2t-1} $, for which we know how to solve the first and second integral but not the third...

Thanks

Hakim
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jreing
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2 Answers2

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Alternatively, if we call our integral $I$ and perform a dummy variable substitution, it all falls out rather nicely! $$I = \int_{0}^{\pi/2} \dfrac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \text{ d}x \ \overset{x \mapsto \frac{\pi}{2} - x}= \ \int_{0}^{\pi/2} \dfrac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \text{ d}x $$ Adding the two integrals together gives $$ 2I = \int_{0}^{\pi/2} \dfrac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \text{ d}x = \int_{0}^{\pi/2} \text{d}x = \dfrac{\pi}{2} \implies \ I = \dfrac{\pi}{4}$$

Khallil
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You mentioned dividing numerator and denominator by $\sqrt{\cos x}$.

$$\int\frac{\sqrt{\tan x}}{\sqrt{\tan x}+1}dx$$

Did you consider the substitution $x=\tan^{-1}u^2$?

$$dx=\frac{2udu}{1+u^4}$$

$$\int\frac{2u^2du}{(u+1)(u^4+1)}$$

I don't remember offhand how to factor $u^4+1$ into quadratics (probably from using the fourth roots of $-1$), but you should be able to use partial fractions from there.

Mike
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  • We started out working on your suggestion, but really our integral is from (0 to $\frac \pi2$ ) - does that affect your substitution somehow or is it still legal doing so? – jreing Oct 19 '13 at 08:32
  • @user1685224 My guess is there are no major complications. We may want to keep in mind, though, the domain of the original function: sine and cosine must be positive. What I've technically done is $u=\sqrt{\tan x}$, hence replacing $\sqrt{\tan x}$ with u. It just makes it a lot easier to see what happens as done above. I'm starting to think things would have been easier had I done $u=-\sqrt{\tan x}$. I may try that out on paper, then expand my answer here. – Mike Oct 19 '13 at 20:09
  • @user1685224 Oops. the other substitution doesn't help. The wrong signs change. You're still left with needing to factor $u^4+1$. – Mike Oct 19 '13 at 20:16
  • An easier way would be to try to decompose $\tfrac{u^2}{u^4+1}$, dividing top and bottom by $u^2$, and using this trick yields $$\begin{align} \frac{1}{\dfrac1{x^2}+x^2}&=\frac{1}{\left(\dfrac{1}x+x\right)^2-2} \&=\dfrac{1} {\left (\dfrac{1}{x}+x+\sqrt{2}\right)\left(\dfrac{1}{x}+x-\sqrt{2}\right)}\&= \dfrac{A}{\dfrac{1}{x}+x+\sqrt{2}}+\dfrac{B}{\dfrac{1}{x}+x-\sqrt{2}}\&=\ldots
    \end{align}$$     – Workaholic Dec 31 '15 at 19:49
  • @Workaholic - For this question Glasser's Master Theorem comes into play as well: https://en.wikipedia.org/wiki/Glasser%27s_master_theorem –  Dec 28 '18 at 09:45