# 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
5 years, 4 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
• Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $\boxed{123}$

Sort by:

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 :)

- 5 years, 4 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??

- 5 years, 4 months ago

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

- 5 years, 4 months ago

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.

- 5 years, 4 months ago

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

- 5 years, 4 months ago

POST! POST! POST! POST! POST! POST! POST! POST!

- 5 years, 4 months ago

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

- 5 years, 4 months ago

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

- 5 years, 4 months ago

Sorry... I just wanted to know how to calculate that limit..

- 5 years, 4 months ago

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

- 5 years, 4 months ago

Oh thanks... Done L'hopital did the job... This was the only point where I struggled in the question but now its done .... Thanks :-)

- 5 years, 4 months ago

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.

- 5 years, 4 months ago

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

- 5 years, 4 months ago

I don't know the differentiation of Beta function.

- 5 years, 4 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.

- 5 years, 4 months ago

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

- 5 years, 4 months ago

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

- 5 years, 4 months ago

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

- 5 years, 4 months ago

Yes sure 8558060484

- 5 years, 4 months ago

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

- 5 years, 4 months ago

Have you got the second question??

- 5 years, 4 months ago

Can you post the solution of It.

- 5 years, 4 months ago

It is a bit long..... Apply integration by parts to evaluate the integral and then apply limit....

- 5 years, 4 months ago

Oka Thanks.

- 5 years, 4 months ago

Welcome.. :-)

- 5 years, 4 months ago

Hey guys please also solve the 3rd Problem.

- 5 years, 4 months ago