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Need Help Integration + Limit

Please Some one provide me the Solution Of This Problem .

And

Please Some one provide me the Solution Of This Problem (2) .

And

Please Some one provide me the Solution Of This Problem (3)

Note by Rishabh Deep Singh
6 months ago

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Hey guys please also solve the 3rd Problem. Rishabh Deep Singh · 6 months ago

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I have also added a new question Please check it Also If anyone can help me solving it. Rishabh Deep Singh · 6 months ago

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@Rishabh Deep Singh Have you got the second question?? Rishabh Cool · 6 months ago

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@Rishabh Cool Can you post the solution of It. Rishabh Deep Singh · 6 months ago

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@Rishabh Deep Singh It is a bit long..... Apply integration by parts to evaluate the integral and then apply limit.... Rishabh Cool · 6 months ago

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@Rishabh Cool Oka Thanks. Rishabh Deep Singh · 6 months ago

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@Rishabh Deep Singh Welcome.. :-) Rishabh Cool · 6 months ago

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It is very easy. Just a matter of 5-6 steps. Use Ramanujan's Master Theorem. I'll provide a solution after 3 April. Aditya Kumar · 6 months ago

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@Aditya Kumar Please send Us solution Now. I need it. Please! Rishabh Deep Singh · 6 months ago

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@Rishabh Deep Singh My parents aren't allowing me to use PC. Shall I send you a pic through whatsapp? Aditya Kumar · 6 months ago

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@Aditya Kumar Yes sure 8558060484 Rishabh Deep Singh · 6 months ago

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@Aditya Kumar I had missed a term that's why I got it as infinity. Aditya Kumar · 6 months ago

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@Aditya Kumar This question was in my Definite Integration Module . Yes it converges. Rishabh Deep Singh · 6 months ago

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I feel its somewhat related to Riemann Sums.But fortunately we have a cool guy in our community: @Rishabh Cool who will post an awesome solution to that problem :) Nihar Mahajan · 6 months ago

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@Nihar Mahajan Unfortunately I had tried that problem many times..... But I'm not able to find from where to start ..... Probably that integration sign is doing most of the harm... There might be a way to get rid of that integration sign.... Probably @Rishabh Deep Singh can help by telling what he has done to crack the problem.... DO you have a hint Nihar?? Rishabh Cool · 6 months ago

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@Rishabh Cool Time to tag seniors: @Pi Han Goh @Ishan Singh @Aditya Kumar Nihar Mahajan · 6 months ago

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@Nihar Mahajan First question.

Step 1) Notice that the integrand is an even function, so \( \displaystyle \int_{-\infty}^{\infty} \cdots = 2 \int_{0}^{\infty} \cdots \).
Step 2) Break the integral into two parts: \(\displaystyle 2 \int_{0}^{\infty} \cdots = 2 \int_{0}^{1} \cdots + 2 \int_{1}^{\infty} \cdots\).
Step 3) For the latter integral, try a substitution of \(y = 1/x\).
Step 4) Combine the 2 integrals into 1. Have you tried GP?
Step 5) Profit.

Second question.

Do you know how to differentiate a Beta function? Differentiate through an integral

Step 1) \(\displaystyle B(a,b) = \dfrac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} = \int_0^{\pi /2} (\sin x)^a (\cos x)^b \, dx \).
Step 2) Differentiiate both sides with respect to \(b\).
Step 3) What is the derivative of a Beta function?
Step 4) Profit. Pi Han Goh · 6 months ago

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@Pi Han Goh I don't know the differentiation of Beta function. Rishabh Deep Singh · 6 months ago

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@Pi Han Goh I think second question cannot be solved by differentiating beta function because limits are not from 0 to pi/2.

Correct me if I am wrong. Harsh Shrivastava · 6 months ago

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@Harsh Shrivastava Limits are indeed from 0 to pi/2. So it's possible. Pi Han Goh · 6 months ago

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@Pi Han Goh 2nd one can also be solved by first evaluating the integral using IBP and then applying limit.... :-) Rishabh Cool · 6 months ago

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@Rishabh Cool POST! POST! POST! POST! POST! POST! POST! POST! Pi Han Goh · 6 months ago

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@Pi Han Goh \[=\lim_{a\to \frac{\pi}{2}^-} (\ln (\cos a)^{\sin a-1})\] how to evaluate this limit ?? Rishabh Cool · 6 months ago

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@Rishabh Cool Why are you posting the actual solution here? Go post it as solution that problem itself. Pi Han Goh · 6 months ago

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@Pi Han Goh Sorry... I just wanted to know how to calculate that limit.. Rishabh Cool · 6 months ago

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@Rishabh Cool Haha don't need to apologize. Have you tried l hospital rule? Pi Han Goh · 6 months ago

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@Pi Han Goh Oh thanks... Done L'hopital did the job... This was the only point where I struggled in the question but now its done .... Thanks :-) Rishabh Cool · 6 months ago

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