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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
k i A Letter to MSM
Arr0w   23
N Sep 19, 2022 by scannose
Greetings.

I have seen many posts talking about commonly asked questions, such as finding the value of $0^0$, $\frac{1}{0}$,$\frac{0}{0}$, $\frac{\infty}{\infty}$, why $0.999...=1$ or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
[list]
[*]Firstly, the case of $0^0$. It is usually regarded that $0^0=1$, not because this works numerically but because it is convenient to define it this way. You will see the convenience of defining other undefined things later on in this post.

[*]What about $\frac{\infty}{\infty}$? The issue here is that $\infty$ isn't even rigorously defined in this expression. What exactly do we mean by $\infty$? Unless the example in question is put in context in a formal manner, then we say that $\frac{\infty}{\infty}$ is meaningless.

[*]What about $\frac{1}{0}$? Suppose that $x=\frac{1}{0}$. Then we would have $x\cdot 0=0=1$, absurd. A more rigorous treatment of the idea is that $\lim_{x\to0}\frac{1}{x}$ does not exist in the first place, although you will see why in a calculus course. So the point is that $\frac{1}{0}$ is undefined.

[*]What about if $0.99999...=1$? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
\begin{align*}
\sum_{n=1}^{\infty} \frac{9}{10^n}&=9\sum_{n=1}^{\infty}\frac{1}{10^n}=9\sum_{n=1}^{\infty}\biggr(\frac{1}{10}\biggr)^n=9\biggr(\frac{\frac{1}{10}}{1-\frac{1}{10}}\biggr)=9\biggr(\frac{\frac{1}{10}}{\frac{9}{10}}\biggr)=9\biggr(\frac{1}{9}\biggr)=\boxed{1}
\end{align*}
[*]What about $\frac{0}{0}$? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
[/list]
Hopefully all of these issues and their corollaries are finally put to rest. Cheers.

2nd EDIT (6/14/22): Since I originally posted this, it has since blown up so I will try to add additional information per the request of users in the thread below.

INDETERMINATE VS UNDEFINED

What makes something indeterminate? As you can see above, there are many things that are indeterminate. While definitions might vary slightly, it is the consensus that the following definition holds: A mathematical expression is be said to be indeterminate if it is not definitively or precisely determined. So how does this make, say, something like $0/0$ indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that $0/0$ is an indeterminate form.

But we need to more specific, this is still ambiguous. An indeterminate form is a mathematical expression involving at most two of $0$, $1$ or $\infty$, obtained by applying the algebraic limit theorem (a theorem in analysis, look this up for details) in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being calculated. This is why it is called indeterminate. Some examples of indeterminate forms are
\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function $f:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}$ given by the mapping $x\mapsto \sqrt{x}$ is undefined for $x<0$. On the other hand, $1/0$ is undefined because dividing by $0$ is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context.

WHEN THE WATERS GET MUDDIED

So with this notion of indeterminate and undefined, things get convoluted. First of all, just because something is indeterminate does not mean it is not undefined. For example $0/0$ is considered both indeterminate and undefined (but in the context of a limit then it is considered in indeterminate form). Additionally, this notion of something being undefined also means that we can define it in some way. To rephrase, this means that technically, we can make something that is undefined to something that is defined as long as we define it. I'll show you what I mean.

One example of making something undefined into something defined is the extended real number line, which we define as
\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let $-\infty\le a\le \infty$ for each $a\in\overline{\mathbb{R}}$ which means that via this order topology each subset has an infimum and supremum and $\overline{\mathbb{R}}$ is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In $\overline{\mathbb{R}}$ it is perfectly OK to say that,
\begin{align*}
a + \infty = \infty + a & = \infty, & a & \neq -\infty \\
a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0).
\end{align*}So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,
\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define $\infty \times 0=0$ as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by $0$ is undefined still! Is there a place where it isn't? Kind of. To do this, we can extend the complex numbers! More formally, we can define this extension as
\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, $\tilde{\infty}$ means complex infinity, since we are in the complex plane now. Here's the catch: division by $0$ is allowed here! In fact, we have
\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]where $\tilde{\infty}/\tilde{\infty}$ and $0/0$ are left undefined. We also have
\begin{align*}
z+\tilde{\infty}=\tilde{\infty}, \forall z\ne -\infty\\
z\times \tilde{\infty}=\tilde{\infty}, \forall z\ne 0
\end{align*}Furthermore, we actually have some nice properties with multiplication that we didn't have before. In $\mathbb{C}^*$ it holds that
\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]but $\tilde{\infty}-\tilde{\infty}$ and $0\times \tilde{\infty}$ are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with $\mathbb{C}^*$, by defining them as
\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]They behave in a similar way to the Riemann Sphere, with division by $0$ also being allowed with the same indeterminate forms (in addition to some other ones).
23 replies
Arr0w
Feb 11, 2022
scannose
Sep 19, 2022
k i Marathon Threads
LauraZed   0
Jul 2, 2019
Due to excessive spam and inappropriate posts, we have locked the Prealgebra and Beginning Algebra threads.

We will either unlock these threads once we've cleaned them up or start new ones, but for now, do not start new marathon threads for these subjects. Any new marathon threads started while this announcement is up will be immediately deleted.
0 replies
LauraZed
Jul 2, 2019
0 replies
k i Basic Forum Rules and Info (Read before posting)
jellymoop   368
N May 16, 2018 by harry1234
f (Reminder: Do not post Alcumus or class homework questions on this forum. Instructions below.) f
Welcome to the Middle School Math Forum! Please take a moment to familiarize yourself with the rules.

Overview:
[list]
[*] When you're posting a new topic with a math problem, give the topic a detailed title that includes the subject of the problem (not just "easy problem" or "nice problem")
[*] Stay on topic and be courteous.
[*] Hide solutions!
[*] If you see an inappropriate post in this forum, simply report the post and a moderator will deal with it. Don't make your own post telling people they're not following the rules - that usually just makes the issue worse.
[*] When you post a question that you need help solving, post what you've attempted so far and not just the question. We are here to learn from each other, not to do your homework. :P
[*] Avoid making posts just to thank someone - you can use the upvote function instead
[*] Don't make a new reply just to repeat yourself or comment on the quality of others' posts; instead, post when you have a new insight or question. You can also edit your post if it's the most recent and you want to add more information.
[*] Avoid bumping old posts.
[*] Use GameBot to post alcumus questions.
[*] If you need general MATHCOUNTS/math competition advice, check out the threads below.
[*] Don't post other users' real names.
[*] Advertisements are not allowed. You can advertise your forum on your profile with a link, on your blog, and on user-created forums that permit forum advertisements.
[/list]

Here are links to more detailed versions of the rules. These are from the older forums, so you can overlook "Classroom math/Competition math only" instructions.
Posting Guidelines
Update on Basic Forum Rules
What belongs on this forum?
How do I write a thorough solution?
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How do I study for mathcounts?
Mathcounts FAQ and resources
Mathcounts and how to learn

As always, if you have any questions, you can PM me or any of the other Middle School Moderators. Once again, if you see spam, it would help a lot if you filed a report instead of responding :)

Marathons!
Relays might be a better way to describe it, but these threads definitely go the distance! One person starts off by posting a problem, and the next person comes up with a solution and a new problem for another user to solve. Here's some of the frequently active marathons running in this forum:
[list][*]Algebra
[*]Prealgebra
[*]Proofs
[*]Factoring
[*]Geometry
[*]Counting & Probability
[*]Number Theory[/list]
Some of these haven't received attention in a while, but these are the main ones for their respective subjects. Rather than starting a new marathon, please give the existing ones a shot first.

You can also view marathons via the Marathon tag.

Think this list is incomplete or needs changes? Let the mods know and we'll take a look.
368 replies
jellymoop
May 8, 2015
harry1234
May 16, 2018
Cost Question
bassamali01   9
N 6 minutes ago by Juno_34
Sorry, I have been struggling with this question so much. It is a simple derivative question as I think it is. Can I get some help on it?
9 replies
bassamali01
Dec 7, 2017
Juno_34
6 minutes ago
Find the marginal profit..
ArmiAldi   1
N 17 minutes ago by Juno_34
Source: can someone help me
The total profit selling x units of books is P(x) = (6x - 7)(9x - 8) .
Find the marginal average profit function?
1 reply
ArmiAldi
Mar 2, 2008
Juno_34
17 minutes ago
2010 Japan MO Finals
parkjungmin   0
38 minutes ago
It's a missing Japanese math competition.

Please solve the problem.

It's difficult.
0 replies
parkjungmin
38 minutes ago
0 replies
ISI UGB 2025 P3
SomeonecoolLovesMaths   6
N an hour ago by Levieee
Source: ISI UGB 2025 P3
Suppose $f : [0,1] \longrightarrow \mathbb{R}$ is differentiable with $f(0) = 0$. If $|f'(x) | \leq f(x)$ for all $x \in [0,1]$, then show that $f(x) = 0$ for all $x$.
6 replies
SomeonecoolLovesMaths
5 hours ago
Levieee
an hour ago
9 What is the best way to learn math???
lovematch13   104
N 2 hours ago by ILOVECATS127
On the contrary, I'm also gonna try to send this to school admins. PLEASE DO NOT TROLL!!!!
104 replies
lovematch13
May 22, 2023
ILOVECATS127
2 hours ago
A mixture $P$ is formed by removing a certain amount of coffee from a coffee jar
Vulch   1
N 2 hours ago by ILOVECATS127
A mixture $P$ is formed by removing a certain amount of coffee from a coffee jar and replacing the same amount with cocoa powder. The same amount is again removed from mixture $P$ and replaced with same amount of cocoa powder to form a new mixture $Q.$ If the ratio of coffee and cocoa in the mixture $Q$ is $16 : 9,$ then the ratio of cocoa in mixture $P$ to that in mixture $Q$ is
1 reply
Vulch
Today at 10:14 AM
ILOVECATS127
2 hours ago
100th post!!!
whwlqkd   1
N 3 hours ago by Pengu14
Hello,I am Seohyung Jo,8th grade student from South Korea. I couldn’t think I write many posts. But now,it is my 100th post!!!! As many people do these kind of posts(like 1000th post),I do this post.

Because this is my 100th post,I will share some problems:make 100

Make 100 with these numbers and +,-,*,/,!,^,nCr,nPr
Level 1(easy):20,25,30,35,40
Level 2(medium):1,3,4,5,7
Level 3(hard):1,2,3,4,5
Level 4(extreme):2,3,6,8,9
Level X:2,3,3,4,4
1 reply
whwlqkd
3 hours ago
Pengu14
3 hours ago
The minor angle between the hours hand and minutes hand of a clock was observed
Vulch   0
Today at 10:06 AM
The minor angle between the hours hand and minutes hand of a clock was observed at $8:48$ am
. The minimum duration, in minutes, after $8.48$ am when this angle increases by $50\%$
is
0 replies
Vulch
Today at 10:06 AM
0 replies
Discover the Future of Coding with WeNextCoder – Your Ultimate Programming Resou
nextcoder   0
Today at 8:13 AM
If you're passionate about technology, programming, or building a career in the digital world, look no further than WeNextCoder.com. This dynamic platform is designed for learners, developers, and tech enthusiasts who want to stay ahead in an ever-evolving industry. Whether you're just starting your journey into coding or you're an experienced developer seeking to expand your skill set, WeNextCoder offers a wide array of resources tailored to every level.

From in-depth tutorials on popular programming languages like Python, JavaScript, and Java to full-stack project guides, real-world coding challenges, and tips for tech interviews, the site provides everything you need to succeed. It also covers the latest trends in web development, artificial intelligence, cloud computing, and more, ensuring you're always up-to-date with what's happening in the tech world.

One of the standout features of WeNextCoder is its beginner-friendly approach, making complex topics easy to understand without watering them down. Plus, the platform's clean interface and organized content make navigation a breeze.

Visit https://wenextcoder.com/ today and take the next step in your programming journey. Empower yourself with the knowledge and tools to build a brighter future in tech.
0 replies
nextcoder
Today at 8:13 AM
0 replies
A container has $40$ liters of milk. Then, $4$ liters are removed from the cont
Vulch   3
N Today at 3:51 AM by evt917
A container has $40$ liters of milk. Then, $4$ liters are removed from the container and replaced with $4$ liters of water. This process of replacing $4$ liters of the liquid in the container with an equal volume of water is continued repeatedly. The smallest number of times of doing this process, after which the volume of milk in the container becomes less than that of water, is
3 replies
Vulch
Yesterday at 10:11 AM
evt917
Today at 3:51 AM
9 middle school olympiads forum ?
kjhgyuio   6
N Today at 2:35 AM by kjhgyuio
There is a high school olympiads forum,so I am thinking why is there no middle school olympiads forum Should i create a middle school olympiads forum ?
here is the link if you are interested ->https://artofproblemsolving.com/community/c4318171_middle_school_olympiads
6 replies
kjhgyuio
Today at 12:28 AM
kjhgyuio
Today at 2:35 AM
MAP Goals
Antoinette14   12
N Yesterday at 11:17 PM by GallopingUnicorn45
What's yall's MAP goals for this spring?
Mine's a 300 (trying to beat my brother's record) but since I'm at a 285 rn, 290+ is more reasonable.
12 replies
Antoinette14
May 8, 2025
GallopingUnicorn45
Yesterday at 11:17 PM
A glass is filled with milk. Two-thirds of its content is poured out and replace
Vulch   4
N Yesterday at 4:23 PM by valisaxieamc
A glass is filled with milk. Two-thirds of its content is poured out and replaced with water. If this process of pouring out two-thirds the content and replacing with water is repeated three more times, then the final ratio of milk to water in the glass, is
4 replies
Vulch
Yesterday at 8:53 AM
valisaxieamc
Yesterday at 4:23 PM
9 MathandAI4Girls!!!
Inaaya   20
N Yesterday at 3:54 PM by Inaaya
How many problems did y'all solve this year?
I clowned and started the pset the week before :oops:
Though I think if i used the time wisely, I could have at least solved 11 of them
ended up with 9 :wallbash_red:
20 replies
Inaaya
May 7, 2025
Inaaya
Yesterday at 3:54 PM
Integration Bee Kaizo
Calcul8er   61
N Today at 6:36 AM by Svyatoslav
Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!
61 replies
Calcul8er
Mar 2, 2025
Svyatoslav
Today at 6:36 AM
Integration Bee Kaizo
G H J
G H BBookmark kLocked kLocked NReply
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Figaro
774 posts
#49
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F.8.
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Calcul8er
21 posts
#50 • 2 Y
Y by aidan0626, franklin2013
Hey all! I thought I'd share the answer sheet (no worked solutions though) so you can verify solutions or potentially gain some insight on how to solve any problems you've been stuck on. I hope you'll enjoy!
Attachments:
Integration_Bee_Kaizo_Answers.pdf (175kb)
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Shikhar_
2 posts
#51
Y by
QF 10
This post has been edited 1 time. Last edited by Shikhar_, Mar 25, 2025, 7:11 PM
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Shikhar_
2 posts
#52
Y by
QF 14
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BaidenMan
1 post
#53
Y by
QR.14.
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franklin2013
290 posts
#54 • 1 Y
Y by aidan0626
This deserves a \bump
As a reminder, these are the integrals that have been solved.
Qualifying Round (FULLY DONE)
Quarterfinals
Semifinals
Finals
This post has been edited 2 times. Last edited by franklin2013, Yesterday at 11:14 PM
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Silver08
461 posts
#56
Y by
Semifinals 14 is botherin me....
Here's my idea:
$$I(n) = \int_0^1 \frac{u^n}{\sqrt{1-u^2}}du = \frac{\sqrt{\pi}\Gamma((n+1)/2)}{2\Gamma(n/2)}$$We want to find $$I^{(3)}(0) = \int_0^{\pi/2}\ln^3(\sin(x))dx.$$But I know nothing about special functions...
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franklin2013
290 posts
#57
Y by
Silver08 wrote:
But I know nothing about special functions...

I knew nothing about calculus and integration until I saw this thread then I attempted to speedrun calc 1 and 2 and succeeded (i guess)
This post has been edited 1 time. Last edited by franklin2013, May 7, 2025, 8:28 PM
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Figaro
774 posts
#59
Y by
@Silver08: regarding SF14, this might interest you:

https://www.youtube.com/watch?v=f9pR30LRnzs&list=PL22w63XsKjqzJpcuD6InKWZXep2L0z1H8&index=6
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franklin2013
290 posts
#61
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I got inspired by Qualifying Round 4's solution, and quickly found the answer to this.

$$I=\int^1_0\left(\left\lfloor\frac{1}{\sqrt[69]{x}}\right\rfloor-420\left\lfloor\frac{1}{420\sqrt[69]{x}}\right\rfloor\right)dx$$Finals 5
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Silver08
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#62
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Figaro wrote:

Ewwwwww, Apery's constant!! >:V
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Calcul8er
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#63 • 1 Y
Y by franklin2013
franklin2013 wrote:
This deserves a \bump
As a reminder, these are the integrals that have been solved.
Qualifying Round (FULLY DONE)
Quarterfinals
Semifinals
Finals

I don't think Semifinals 4 has been answered here. The solution posted had some issues.
This post has been edited 1 time. Last edited by Calcul8er, Yesterday at 11:13 PM
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franklin2013
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#64
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Ok! I've updated my post
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Svyatoslav
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#65
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SF 8 $\qquad I = \int_{-\infty}^\infty \frac{\cos(x) \cos \sqrt{x^2+2}}{x^2+1} dx$
We have to be accurate when handling branch points. Using the symmetry of the integrand,
$$I= \int_{-\infty}^\infty \frac{\cos(x +\sqrt{x^2+2})}{x^2+1} \,dx\overset{x=\sqrt2\sinh t}{=}\sqrt2\int_{-\infty}^\infty\frac{\cos\big(\sqrt2e^t\big)\cosh t}{2\sinh^2t+1}dt=\sqrt2\int_{-\infty}^\infty\frac{\cos\big(\sqrt2e^t\big)(e^t+e^{-t})}{e^{2t}+e^{-2t}}dt$$$$\overset{e^t=x}{=}\frac1{\sqrt2}\int_{-\infty}^\infty\frac{\cos(\sqrt2x)(x^2+1)}{x^4+1}dx=\frac1{\sqrt2}\Re\,\int_{-\infty}^\infty\frac{e^{i\sqrt2x}(x^2+1)}{x^4+1}dx$$Closing the contour in the upper half-plane, we have inside the contour two simple poles - at $x=e^{\frac{\pi i}4}$ and $x=e^{\frac{3\pi i}4}$.
The residues evaluation is straightforward and gives
$$I=\frac1{\sqrt2}\Re\,2\pi i\left(\frac{e^{i\sqrt2e^{\frac{\pi i}4}}(1+i)}{(e^{\frac{\pi i}4}-e^{\frac{3\pi i}4})(e^{\frac{\pi i}4}-e^{\frac{5\pi i}4})(e^{\frac{\pi i}4}-e^{\frac{7\pi i}4})}+\frac{e^{i\sqrt2e^{\frac{3\pi i}4}}(1-i)}{(e^{\frac{3\pi i}4}-e^{\frac{\pi i}4})(e^{\frac{3\pi i}4}-e^{\frac{5\pi i}4})(e^{\frac{3\pi i}4}-e^{\frac{7\pi i}4})}\right)=\frac\pi e\cos(1)=0.62444...$$
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Svyatoslav
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Regarding F8 - we can find a more general solution in a simpler way
Let's take $\lambda\geqslant1$, $a>b>0$ and consider
$$I(\lambda, a, b)=\int_{-\infty}^\infty\frac{\cos(\lambda x)\cos\sqrt{x^2+a^2}}{x^2+b^2}dx=\Re\int_{-\infty}^\infty\frac{e^{\lambda ix}\cos\sqrt{x^2+a^2}}{x^2+b^2}dx$$On the one hand, the function $f(x)=\cos\sqrt{x^2+a^2}$ is analytical in all complex plane (because its Taylor's series is analytical and valid in all complex plane). On the other hand, in the upper half-plane at $\lambda\geqslant1$ the integrand is bounded by $\frac{\text{const}}{x^2+b^2}$
Closing the contour in the upper-half-plane and taking the residue at $x=ib$
$$\boxed{\,\,I(\lambda, a, b)=\frac\pi be^{-\lambda b}\cos\sqrt{a^2-b^2}\,\,}$$Taking $\lambda=1$, we get the value of the desired integral
$$I(a,b)=\frac\pi be^{-b}\cos\sqrt{a^2-b^2}$$_______________________
At $b>a$ one can show via direct evaluation that
$$I(a,b)=\frac\pi be^{-b}\cosh\sqrt{b^2-a^2},\quad b>a>0$$or the same result can be obtained from the formula above, just using $\cos(ic)=\cosh c$
This post has been edited 1 time. Last edited by Svyatoslav, Today at 8:52 AM
Reason: addendum
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