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I guess thesame problem can be solved by taking t as $$\sin ^2 x$$ and using waali's formula so what is the use of gamma function when there is a simpler approach ?

Note by Athul Arun A
2 years, 5 months ago

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First thing, which I would like to mention is that, can you please share your complete approach?

Secondly, it isn't that Waali's formula is a simpler approach. One may find implementing Gamma Function to solve the problem, an easier approach. We cannot be so absolute in saying that Waali's Formula is a simpler approach. After all, it all depends on your point of view. (Think of a situation in which the reader is not familiar with Waali's formula or maybe he/she is not able to observe that the direct result follows by Waali's Formula)

Thirdly, Waali's Integrals have an abstract relationship with Beta and Gamma Functions as follows

$W_n = \frac{1}{2}B\left(n+\frac{1}{2},\frac{1}{2}\right) = \frac{\sqrt{\pi}}{2}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2}+1)}$

where

$W_n = \int_0^{\pi/2} \sin^n x \mathrm{d}x = \int_0^{\pi/2} \cos^n x \mathrm{d}x$

The above results follow from the substitution $$t \rightarrow \sin^2 x$$ in beta function, about which you have mentioned in your note.

Also, as you wrote that "what is the use of gamma function", they not only find use in solving such kind of problems but also in solving series and in many other areas. You may read the wiki on Gamma Functions for such specific examples. And yes, I would urge that please provide us with your complete approach or tell us about your ideas so that we can have a more detailed discussion.

I hope you find the answer for your question in my above comment.

Thanks. · 2 years, 5 months ago

Thank you.... In our book without showing abstract for waali's integral the substitution was directly used with formula so , I assumed it to be rather easier as I don't have much idea about gamma functions. · 2 years, 5 months ago

Yeah, I understand, JEE Mathematics course is very much limited. · 2 years, 5 months ago