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
3 years, 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 - 3 years, 6 months ago

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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 Jain - 3 years, 6 months ago

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Time to tag seniors: @Pi Han Goh @Ishan Singh @Aditya Kumar

Nihar Mahajan - 3 years, 6 months ago

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@Nihar Mahajan First question.

Step 1) Notice that the integrand is an even function, so =20 \displaystyle \int_{-\infty}^{\infty} \cdots = 2 \int_{0}^{\infty} \cdots .
Step 2) Break the integral into two parts: 20=201+21\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/xy = 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) B(a,b)=Γ(a)Γ(b)Γ(a+b)=0π/2(sinx)a(cosx)bdx\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 bb.
Step 3) What is the derivative of a Beta function?
Step 4) Profit.

Pi Han Goh - 3 years, 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 Jain - 3 years, 6 months ago

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@Rishabh Jain POST! POST! POST! POST! POST! POST! POST! POST!

Pi Han Goh - 3 years, 6 months ago

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@Pi Han Goh =limaπ2(ln(cosa)sina1)=\lim_{a\to \frac{\pi}{2}^-} (\ln (\cos a)^{\sin a-1}) how to evaluate this limit ??

Rishabh Jain - 3 years, 6 months ago

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@Rishabh Jain Why are you posting the actual solution here? Go post it as solution that problem itself.

Pi Han Goh - 3 years, 6 months ago

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@Pi Han Goh Sorry... I just wanted to know how to calculate that limit..

Rishabh Jain - 3 years, 6 months ago

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@Rishabh Jain Haha don't need to apologize. Have you tried l hospital rule?

Pi Han Goh - 3 years, 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 Jain - 3 years, 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 - 3 years, 6 months ago

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@Harsh Shrivastava Limits are indeed from 0 to pi/2. So it's possible.

Pi Han Goh - 3 years, 6 months ago

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@Pi Han Goh I don't know the differentiation of Beta function.

Rishabh Deep Singh - 3 years, 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 - 3 years, 6 months ago

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I had missed a term that's why I got it as infinity.

Aditya Kumar - 3 years, 6 months ago

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Please send Us solution Now. I need it. Please!

Rishabh Deep Singh - 3 years, 6 months ago

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My parents aren't allowing me to use PC. Shall I send you a pic through whatsapp?

Aditya Kumar - 3 years, 6 months ago

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@Aditya Kumar Yes sure 8558060484

Rishabh Deep Singh - 3 years, 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 - 3 years, 6 months ago

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Have you got the second question??

Rishabh Jain - 3 years, 6 months ago

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Can you post the solution of It.

Rishabh Deep Singh - 3 years, 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 Jain - 3 years, 6 months ago

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@Rishabh Jain Oka Thanks.

Rishabh Deep Singh - 3 years, 6 months ago

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@Rishabh Deep Singh Welcome.. :-)

Rishabh Jain - 3 years, 6 months ago

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Hey guys please also solve the 3rd Problem.

Rishabh Deep Singh - 3 years, 6 months ago

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