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)

#Calculus

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 - 5 years, 2 months ago

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 - 5 years, 2 months ago

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

Nihar Mahajan - 5 years, 2 months ago

@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 - 5 years, 2 months ago

@Pi Han Goh – 2nd one can also be solved by first evaluating the integral using IBP and then applying limit.... :-)

Rishabh Jain - 5 years, 2 months ago

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

Pi Han Goh - 5 years, 2 months ago

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

Rishabh Jain - 5 years, 2 months ago

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

Pi Han Goh - 5 years, 2 months ago

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

Rishabh Jain - 5 years, 2 months ago

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

Pi Han Goh - 5 years, 2 months ago

@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 - 5 years, 2 months ago

@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 - 5 years, 2 months ago

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

Pi Han Goh - 5 years, 2 months ago

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

Rishabh Deep Singh - 5 years, 2 months ago

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 - 5 years, 2 months ago

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

Aditya Kumar - 5 years, 2 months ago

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

Rishabh Deep Singh - 5 years, 2 months ago

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

Aditya Kumar - 5 years, 2 months ago

@Aditya Kumar – Yes sure 8558060484

Rishabh Deep Singh - 5 years, 2 months ago

I have also added a new question Please check it Also If anyone can help me solving it.

Rishabh Deep Singh - 5 years, 2 months ago

Have you got the second question??

Rishabh Jain - 5 years, 2 months ago

Can you post the solution of It.

Rishabh Deep Singh - 5 years, 2 months ago

@Rishabh Deep Singh – It is a bit long..... Apply integration by parts to evaluate the integral and then apply limit....

Rishabh Jain - 5 years, 2 months ago

@Rishabh Jain – Oka Thanks.

Rishabh Deep Singh - 5 years, 2 months ago

@Rishabh Deep Singh – Welcome.. :-)

Rishabh Jain - 5 years, 2 months ago

Hey guys please also solve the 3rd Problem.

Rishabh Deep Singh - 5 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$