#Integrals

[callmebliss]

Y’ever read something and have understanding that has eluded you interminably suddenly stop, curl up, and snuggle neatly into a fold in your brain because a new way opened to it?

[kneadingmaths]

[secondwheel]

The  |  stands for the absolute value of an integer. In this case, it makes anything positive. 

[math-imagines]

Black Doom Shows You How to Do Integration by Substitution

Gender Neutral Reader, Black Doom & Reader or Black Doom/Reader, up to interpretation

Being in space, you would have assumed that you wouldn’t have to worry about things like math and calculus. It wasn’t like you assumed it wouldn’t be involved in space travel, but you assumed that you wouldn’t have to deal with it with this exact space trip, considering you were abducted by aliens and all. You’d assumed that you were going to be eaten, like everyone else who’d been captured. Yet you and a very small number of others hadn’t yet and were instead being put through a number of tests. A few were physical in some way, but most had tested your intelligence. You couldn’t help but feel like a rat in a maze, being tested on how fast we could change our behavior after being electrocuted. Were these aliens, the Black Arms as you now knew they were called, studying you? If they were, then what for?

This specific test was a test of your ability to do calculations. Vaguely you thought back on that comic that was rather popular a while ago, about someone being forced to do algebraic equations to get into heaven, and couldn’t help but feel that this was something like that. The first few questions weren’t too hard for you, you’d never been bad at math. But when you got into the calculus section, you started to get nervous. You’d only learned so much calculus before you’d been, well, abducted by aliens. And one part in particular always go to you. Integrals in general just hadn’t clicked with you yet. You knew how to do it, but you couldn’t always do it right. It didn’t help that the massive ruler of the Black Arms, Black Doom, was watching everyone testing. You were sure it couldn’t be, but it was almost like he was watching you more than the others.

The next question was about just one of these. Of course it was. It asked what the indefinite integral was of the cube root of seven minus tangent t, all multiplied by secant squared of t. Nothing seemed to be going your way. Why did they even to know if you could do integrals anyway? Surely, they were just going to eat you in the end anyway! Why did they need you to work alone too? Wouldn’t this test be just as effective if you could all work together? You tried to solve the problem anyway, if you only could get a little help…

The floating Black Doom started to move through the space, and to your horror, he came towards you! Had you displeased him in some way? You were quite certain this was when you would be eaten, when he said in his low voice, “You struggle. Why do you persist?”

You glanced around, hoping there would be someone else nearby to respond, but no one did, so you answered nervously, “I don’t know how to do this bit. I hadn’t learned it in class yet. I think I can get it though, if someone…” You paused for a moment, considering whether or not this was a test too. Should you say it? Was that against the rules? No one else spoke, and some others clearly struggled too.

He seemed to read your mind as he let out a low laugh. “It is no shame to require assistance,” he said, “The Black Arms are a hive mind, there is a constant cooperation and collaboration between us. I’ve not seen one of your kind request assistance, not in a non-life threating situation. It takes intelligence to be capable of learning and wisdom to be willing to ask questions. Certainly, you are a promising candidate. Come with me, I will show you how to solve problems such as these.”

…what? Black Doom, ruler of the Black Arms, probably a demi-god, destroyer of worlds, was offering to teach you how to do calculus? Called you intelligent and wise? This couldn’t be real. Shakely, you stood and followed the towering alien out of the room. You still weren’t sure you weren’t going to be eaten, but you followed obediently. And how did he know what you were going to ask for help? Could he read your mind?

“Yes,” he said, though you hadn’t spoken a word. You walked with him down a long corridor now. “We’ve partially attuned you and your fellows to the hive mind. You cannot connect to it, but we can examine your thought processes through it. We use it to determine how well you complete the tests to reveal which of you are most suitable.”

“How are you able to do that?” you asked, knowing there’s no point in leaving it unsaid now. You tried not to think about what “suitable” referred to.

“It’s in your food. A psychokinetically active additive makes your small minds more open to us. We need you all alive for now, so you must eat to live.”

“So, it’s like the radioactive compounds in PET scans, but to see our thoughts instead of our brains?”

He glanced at you, giving a judgmental gaze. “They do not have those in the rest of the civilized universe. Most do not put radioactive compounds in their bodies willingly.”

“Oh.” You walked in silence down the long corridor, until you followed him into an antechamber with a giant tablelike structure in the center.

“I will show you how to complete problems like these,” he declared, moving to one side of the table and indicating that you should join him by scaling a massive seat that seemed more like a monolith than a chair. You managed, with a reasonable amount of effort to climb on top of it. “Good,” he said as you gazed up at him, “You struggled with this.” You nodded as he indicated the problem in question.

“It simply a matter of substitution,” he said, using a massive claw to etch the problem on the table.

“You must simply replace one part of the integral with a separate variable, making it easier to integrate.”You weren’t quite sure you knew what he meant, but you nodded along as he etched a separate variable, u, into the table.

He continued. “It would be more difficult to integrate the cube root of 7 tangent t than of a single variable, such as u. To integrate with respect to u, you must also transform dt to du.” He then wrote a derivative equation:

“Why would we take the derivative of u?”, you asked, confused at the seemingly irrelevant step.

“Have patience,” Black Doom snapped, then continued, “If you’d not interrupted, I would have been able to inform you that the derivative of u is taken to isolate du and dt, and their equivalent portions of the equations.” As he took the derivative of 7 minus 7 tangent t, it made more sense; the majority of the derivative was the factor outside of the radical.

“Multiplying both sides by dt isolates du and dt, making it possible to replace secant squared t and dt with an equivalent expression in that makes the integral possible to integrate in terms of u rather than t.”

When he lay it out before you, it seemed far simpler than you thought it would be. “So, you can take a simpler integral rather than a more complicated one by replacing parts of it with equivalent and simpler variables?”

“Correct,” If Black Doom had a mouth, you could have sworn smiled at you. “Your species’ capacity to learn so rapidly is something we valued in you. If I were to name a positive characteristic in your inferior species, it would be that.”

You gave a weak smile in response, somewhat concerned by his phrasing. You watched him write the edited integral.

This integral was far more reasonable to deal with, a simple matter really. You etched the integrated solution on the table in your smaller hand writing with your writing utensil, something between a pencil and marker.

Black Doom nodded approvingly. “There is but one more step,” he said, “to replace u with what it was originally substituted for.”

“That’s the solution then?” you confirmed.

“Indeed,” he replied. “You are certainly a most suitable candidate. Very adaptable and obedient. I shall keep you in mind for later. Now return to your fellows. Do not bother with completing more of this test, your progress has been deemed satisfactory.”

You nodded and rushed towards the exit, then paused for a moment. “Thank you,” you said, trying not to make your voice shake.

Black Doom looked at you curiously for a moment, after which he said, “As you should. Now begone!”

As you turned and returned to the first chamber you were in, you gave the test you’d completed in a pile of other completed tests and returned to a group of your fellow abductees.

“Oh my chaos!” one gasped as you returned, “You’re alive! We thought for sure you got eaten!”

You shook your head. “No,” you said, “Actually, Black Doom explained calculus to me.”

“You’re clowning? You’re not clowning? I sense clowns.”

“No, I’m serious.” You turned away. “I think there’s something more going on here, though,” you whispered, “I’ve told you before I think they’re studying us, right?”

The others nodded, and some of the Black Arms nearby almost imperceptibly tensed, shifting.

You glanced around, “I think these tests are for more than just study,” you said softly, “I think there’s a bigger goal here. We’re candidates for something. These are just the preliminary tests to determine which of us is best for the real experiment.”

The rest of the group recoiled. “Surely not?” one chuckled nervously. “I mean, the aliens wouldn’t have told you their real plan, right?”

You opened your mouth to double down, but one of the Black Arms jerked their head and reached for a phaser at the edge of your vision. “You’re probably right,” you said, keeping your eye on the alien, “just a theory.”

The group visibly relaxed and so did the surrounding Black Arms. You didn’t though. What had happened still didn’t feel real to you. But as much as you knew something else was going on, you pushed it to the back of your mind, in a place you hoped the Black Arms couldn’t find.

***

Author's Note: Hope you enjoyed and/or learned some calculus from this! Again, if you have any character requests for any fandom or any math topic, just ask. I feel completely deranged for writing this, but I sure did review that topic! I think the plot's not terrible either. Until next time!

[nerdymemes]

[horatiocomehome]

integration. it's just simply not done anymore! no one is doing integration. it's passe. it's out. and that's a good thing! people are saying this!

differentiation is and always has been, simply objectively better. there is NOTHING, absolutely not a single problem, where integration achieves something that could not be done by simply looking at a problem and knowing in your heart what differentiation was done to obtain it.

it's true. and we all need to get with the times. thank you.

[callmehyungoppaokay]

What it feels like to do sums of matrices and determinants

What it feels like to do sums of integrals and derivatives

[giantjupiter]

I just realized that there's an amazing book on calculus which people can use to learn it. Some have probably heard it from the YT channel Mathologer. It's out there on the web, for FREE.

Here it is:

How good is this book? Well, check this out:

(idk if the image is blurry though... but you get the point.)

Happy calculus!

[er-cryptid]

Trigonometry Integrals

∫ sin u du  =  -cos u + C

∫ cos u du  =  sin u + C

∫ tan u du  =  -ln |cos u| + C

∫ cot u du  =  ln |sin u| + C

∫ sec u du  =  ln |sec u + tan u| + C

∫ csc u du  =  -ln |csc u + cot u| + C

∫ sec^2 u du  =  tan u + C

∫ csc^2 u du  =  -cot u + C

∫ sec u tan u du  =  sec u + C

∫ csc u cot u du  =  -csc u + C

[bpomo]

arthur berzinsh

[ratsoupee]

Every once in a while, when the art block happens, i simply revert to my true form: Undertale Fanartist

[pinyonrice]

I DUNNO! IT JUST SOUNDS LIKE BEING ALIVE TO ME!

Edit, Jan 2025: hi if you're going to ask for me to turn reblogs on or ask why they're off I have a suggestion: don't! They're off because I want them to be and that's that. I'm tired of being nice about this stupid fucking post. Don't tell me how nothing matters because we're going to die on a post about living, or make weird comments on a post about my body, or insert your weird "I am a god" soapbox spiel into my deeply personal post. This art is not about you, and it never fucking will be. The people on this website don't know how to interact with other people normally.

[math-imagines]

Sigh. Orbot I could've figured that one by myself + should've passed that one already. Thank you though.

[welcometoqueer]

I haven’t really seen any of the more recent U.S. election news hitting tumblr yet so here’s some updates (now edited with sources added):

There’s evidence of Trump cheating and interfering with the election.

Possible Russian interference.

Mail-in ballots are not being counted or “recognized” in multiple (notably swing) states.

30+ bomb threats were called in and shut down polling stations on Election Day.

20+ million votes are still unaccounted for, and that’s just to have the same voter turnout as 2020.

There was record voter turnout and new/first-time voter registration this year. We definitely should be well over the turnout in 2020.

U.S. citizens are using this site to demand, not only a recount, but a complete investigation into election fraud and interference for the reasons stated above:

Here is what I submitted as an example:

An investigation for election interference and fraud is required. We desperately need a recount or even a revote. The American people deserve the right to a free and fair election. There has been evidence unveiled of Trump cheating and committing election fraud which is illegal. There is some evidence of possible Russian interference. At least 30+ bomb threats were called in to polling places. Multiple, notably swing states, have ballots unaccounted for and voting machines not registering votes. Ballots and ballot boxes were tampered with and burned. Over 20 million votes that we know of are unaccounted for. With record turnout and new voter registration this year, there should be no possibility that there are less votes than even in the 2020 election.

Sources (working on finding more links but if anyone wants to add info, it’s appreciated):

FBI addressing Russian interference and bomb threats:

Emails released by Rachael Bellis (private account, can’t share original tweet) confirming Trump committing election fraud:

Pennsylvania's Centre County officials say they are working with their ballot scanner vendor to figure out why the county's mail-in ballot data is "not being recognized when uploaded to the elections software:”

Wisconsin recount:

[ID:

Multiple screenshots and images.

The first is a screenshot with a link and information for contacting the White House directly regarding election fraud. The instructions include choosing to leave a comment to President Joe Biden directly and to select election security as the reason.

The screenshot then instructs people to include any or all of the following information in a paragraph as a comment to the president:

32 fake bomb threats were called into Democratic leaning poll places, rendering polling places closed for at least an hour.

A lot of people reporting their ballots were not counted for various reasons.

This all occurred in swing states.

This is too coincidental that these things happen and swing in his favor after months of hinting at foul play.

Directly state that an investigation for tampering, interference, fraud is required, not just a recount.

The second image is from the FBI Twitter account that reads:

The FBI is aware of bomb threats to polling locations in several states, many of which appear to originate from Russian email domains. None of the threats have been determined to be credible thus far. https://t.co/j3YfajVK1m — FBI (@FBI) November 5, 2024

The next four Gmail screenshots of an email sent to Rachael Bellis from Chris T. Spackman that read together as follows:

Dear BELLIS, RACHAEL E., The Dauphin County Board of Elections received a challenge to your absentee ballot you applied for in the November 5, 2024 General Election. The challenge argues that a provision of the Pennsylvania Election Code takes precedence over the federal Uniformed and Overseas Citizens Absentee Voting Act (UOCAVA), which requires states and counties to permit U.S. citizens who move overseas to vote by absentee ballot for federal offices based on their last U.S. residential address.

The full text of the challenge that was filed appears below this email.

You may respond to the challenge in any of the following ways:

1. Call the Bureau of Registration and Election at (717) 780-6360;

2. Email a statement to the Bureau at Election election@dauphincounty.gov. Any statement you submit regarding the period during which you lived in Dauphin County, any family or connections that you still have here, and why you are now residing abroad would be read into the record.

3. Appear in person at a Board of Elections hearing scheduled for Friday, November 8 at a time to be determined in the Commissioners Public Hearing Room, 4th floor of Dauphin County Administration Building, 2 S 20d St, Harrisburg, PA 17111. The meeting is also likely to be livestreamed on Facebook on the Dauphin County channel.

Sincerely,

Christopher T Spackman

TEXT OF CHALLENGE BEGINS

Dear Dauphin County Board of Elections,

I am submitting this challenge to an absentee ballot application pursuant to 25 Pa. Stat.

3146.8(f).

25 Pa. Stat. 3146.8(f) Any person challenging an application for an absentee ballot, an absentee ballot, an application for a mail-in ballot or a mail-in ballot for any of the reasons provided in this act shall deposit the sum of ten dollars ($10.00) in cash with the county board, which sum shall only be refunded if the challenge is sustained or if the challenge is withdrawn within five (5) days after the primary or election. If the challenge is dismissed by any lawful order then the deposit shall be forfeited. The county board shall deposit all deposit money in the general fund of the…

The rest of the forwarded email is cut off.

The last image is a screenshot of the official statement from the Centre County, Pennsylvania Board of Commissioners released on November 6, 2024 that states:

Centre County Working with Ballot Scanner Vendor to Export Election Results.

(Bellefonte, PA) -Centre County Elections Office is working continuously to provide mail-in ballot data in order to post unofficial results.

To this point, all ballots have been scanned, including all mail-in ballots.

Centre County's Election team and IT team have identified that the data are successfully being exported from the mail-in ballot scanners, but that the data is not being recognized when uploaded to the elections software.

Centre County's Administrator, John Franek, Jr. stated, "We have not stopped working, and we will continue to work until unofficial results are posted and reported to the Pennsylvania Department of State."

As a next step, Centre County has begun working with the equipment vendor to adjust configurations to make the two systems-the mail-in ballot scanner and the elections software where data are uploaded -compatible with one another.

We will provide updates as we make progress.

/end ID]

[robby94ls]

Saw a sad little teg the other day. 😕😕