Integral Contest

I am always amazed to those who can answer my recent OPs. I wonder, what kind of approach did they use to solve my OPs like:

They have no solutions yet, so feel free to add them. ☜(ˆ▿ˆc)

Now, I hold an unofficial small contest among users here. Just for having fun guys. I hope Brilliant.org users can solve it and provide us a full answer (step-by-step solution) without the aid of machines like Wolfram Alpha, Mathematica, Maple, etc. Here is the problem.

How does one prove the following integral?

$\Large\color{#3D99F6}{\int_0^{\Large\frac{\pi}{2}} \ln\left(\,\frac{\ln^2\sin\theta}{\pi^2+\ln^2\sin\theta}\right)\,\frac{\ln\cos\theta}{\tan\theta}\,d\theta=\frac{\pi^2}{4}}$

This problem is also posted on Mathematics StackExchange as ongoing contest and Quora. You may also answer it there.

I will follow the one who can give a brilliant or nice answer (based on my personal choice or the highest voted answer) to this problem as a token of my appreciation for your answer. What else can you get? Almost nothing, but at least you can brag yourself about how good you are at integration topic among other users here and of course your friends.

Please make a substantial comment on this note. If possible, the answer of this problem only. Also, please do not ask this problem on other sites. Do not cheat and do it by yourself, be honest. Okay, happy problem solving and good luck guys! (>‿◠)✌

Anastasiya Romanova

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

example link

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    # 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"

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}

Easy Math Editor

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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
paragraph 1 paragraph 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"

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

Nice problem, I enjoy it very much during my spare time. BTW, can you contact me Valentina or Anastasiya? I think you know how to reach me. I'll send you my answer to this problem. It's not $\LaTeX$ -formatted anyway, just some of snapshot pictures. I'm really busy these days, so I don't have time to type the answer. :)

Tunk-Fey Ariawan - 6 years, 7 months ago

Hi Mr. @Tunk-Fey Ariawan . Long time no see. Where have you been? Anyway, I also posted this problem in Math SE, so could you post your answer there after I give bounty to that OP? I really want you to participate in my contest there. Since it's currently put as off topic [on hold], could you please vote to reopen? Don't forget to upvote too. ≧◠‿◠≦✌

Anastasiya Romanova - 6 years, 7 months ago

Test

John Young - 4 years, 11 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}`	$\boxed{123}$