Who's up to the challenge? 71

Calculus Level 5

$\int _{ 0 }^{ \pi /2 }{ \sqrt { \sin { x } } \, dx } =\dfrac { \sqrt { A } }{ \sqrt { \pi } } \left(\Gamma \left( \frac { C }{ D } \right)\right)^B$

The equation above holds for positive integers $A,B,C$ and $D$ with $C,D$ coprime. Find $A+B+C+D$ .

Notation : $\Gamma(\cdot)$ denotes the Gamma function .

The answer is 11.

1 solution

Aditya Narayan Sharma
Jun 8, 2016

$\displaystyle \mathfrak{L}(a)= \int_{0}^{\frac{\pi}{2}} (\sin x)^a = \frac{2\sqrt{\pi}\Gamma(\frac{a+1}{2})}{\Gamma(\frac{a+2}{2})}$ by definition of Beta Function

$\displaystyle \mathfrak{L}(\frac{1}{2}) = \frac{2\sqrt{\pi}\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}$

Using Reflection Identity $\displaystyle \Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin(\pi x)}$ for $0<x<1$ ,

$\displaystyle \int_{0}^{\frac{\pi}{2}} \sqrt{\sin x} = \frac{\sqrt{2}}{\sqrt{\pi}} \Gamma^2(\frac{3}{4})$ , Thus the answer $\boxed{2+2+3+4=11}$

It's much easier if you just apply the Beta function formula .

Pi Han Goh - 5 years ago

Yes the beta will give a one line solution infact, I just made a generalization in terms of gamma

Aditya Narayan Sharma - 5 years ago

Then you have not proved your first line.

Pi Han Goh - 5 years ago

@Pi Han Goh – Ah ok , you're ri8. I am mentioning the use of beta there as proving the beta function will make the solution lengthy.

Aditya Narayan Sharma - 5 years ago

Who's up to the challenge? 71

The answer is 11.

1 solution

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