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.

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold

- bulleted - list

bulleted

list

1. numbered 2. list

numbered

list

Note: you must add a full line of space before and after lists for them to show up correctly

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 :)

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??

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.

Easy Math Editor

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:

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in`\(`

...`\)`

or`\[`

...`\]`

to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestI 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 :)

Log in to reply

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??

Log in to reply

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

Log in to reply

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.

Log in to reply

Log in to reply

Log in to reply

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

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Correct me if I am wrong.

Log in to reply

Log in to reply

Log in to reply

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

Log in to reply

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

Log in to reply

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

Log in to reply

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

Log in to reply

Log in to reply

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

Log in to reply

Have you got the second question??

Log in to reply

Can you post the solution of It.

Log in to reply

Log in to reply

Log in to reply

Log in to reply

Hey guys please also solve the 3rd Problem.

Log in to reply