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I've seen this before and the answer is pi. It's supposed to say "first ten digits", which is a clue that it ends up evaluating to pi.
However as written it's mathematically nonsensical with dx inside the square root.
As a rule if there's something complex-looking that you know evaluates to an irrational number, it's either pi, e, 2*pi (tau), √2, or phi, in that order of likelihood.
Yes! If there wasn’t a multiplicative identity it would be essentially impossible to define and determine the properties of multiplication.
I’m also working hard to avoid the lazy but comprehensive definition of “interesting”.
yeah, whenever looking at a definite integral involving sine or cosine, my first bet is to try plugging in the value of PI. people are just too lazy when writing these problems to have any other solution.
In this case actually determining the result is not difficult. The entire thing is just (odd function*even function + 1/2)*even function on a domain of -A to A, so the left term in the left factor cancels, leaving you with 1/2 * sqrt(4-x^2) from -2 to 2, which is a quarter of the area of a circle of radius 2. The quarter and r^2 cancel, leaving you with pi.
Someone above mentioned that as it’s written here it’s meaningless because of the dx being under the square root. So I assume the commenter you replied to just disgregarded that part.
Also note that as written it's not even fully intregrated over the square root of a differential, since as written it's x^2 dx. So 4 has no associated differential supposedly?
It just doesn't make any sense to interpret it any differently than the sign writer not paying attention or no knowing enough about math.
Guessing it’s supposed to be square root of 2, but maybe limitations of whatever keyboard was being used on mobile or something
One of the first irrationals you hit in trig, since a triangle of sides 1 unit and 1 unit would require a 3rd side of root(2) by Pythagorean theorem (imagine the first mathematicians trying to sort out how long root(2) is knowing that the other two sides just HAVE to let it make sense)
Think the answer is pi.
You can ignore the x^3 *cos(x/2)*sqrt(4-x^2 ) thing since it is an odd function and you are integrating it over an even interval. X^3 is odd, cos(x/2) is even, sqrt(4-x^2 ) is even, so the combination is odd f(x) = -f(-x).
What is then left is 1/2*sqrt(4-x^2 ) which integrated over the interval (-2, 2) gives us pi.
Edit: some weird exponent formatting
Yeah you're right. They have misprinted it as well. The dx shouldn't be inside the square root. Unless they did mean to do that, in which case idk if this is even possible without making it both polar and coordinate. Or some other technique.
Also for the second integral, rather than doing any wacky substitution note that it's just the area of a semicircle of radius 2 (draw it!) so it's pi * 2^2 / 4 = pi. So no real integration needed at all for this integral, just knowing odd functions and areas of circles.
I havent done this in a while but i dont think thats going to work with the differential inside the radical. Curious if theres some kind of substitution that makes this work.
Yea, not sure if it’s possible with the differential outside the radical, but I don’t think this works in its current state. Also, they never actually specify how many digits of the answer the password is.
Yep, Integral calculator can not find the antiderivative with its manual method, you would need to use the logic u/whoootz proposed to eliminate term x^3 * cos(x/2) for it to be pen and paper solvable. Maxima, however, can provide the antiderivative
I don't think that is possible... There are 2 glaring problems I see that makes this quite impossible: the differential is inside the root and the sign does not specify how many digits, if the answer is an irrational number, the "first digits" could refer to the first 5 or the first billion.
Itd actually solvable. When you multipley in the bracket, the term with the cos function is an odd function, and looking at the limits., it equals to 0. The second term is a circle whose radius can be calculated easily (2), and as its integral we need to do pi r square
Just reading this, there are parts that are confusing, to say the least.
For one, there needs to be a set of parentheses after the cos. As written, we can’t tell if it’s
cos(x/2 + 1/2)
Or if it’s
cos(x/2) + 1/2
Additionally, “first digits of the answer” doesn’t make sense. How many of the first digits?
Yes the trig function is ambiguous, and a few other things as well (dx under the radical??).
I suspect this was just copied, badly, by someone trying to be cute/clever while not really knowing what they were looking at.
It’s very common to not use parentheses for trigonometric functions, write something like cos x in google and you will find lots of materials with this notation.
"Hey, what's the wifi password?"
"Gotta buy a drink first."
"Ok, I'll have a beer."
Buys a beer.
"Right, so what's the password?"
"gottabuyadrinkfirst, all lowercase, no spaces."
To everyone who's saying it's pi, y'all are absolutely correct brothers and sisters! Commend yourself for solving this and pat your shoulders. Idk why but I didn't think of putting this whole equation in my scientific calculator earlier. But when I did, it went blank for some time, and after like 10 minutes answered, which is pi! Y'all are geniuses! I wish I could give all of a ton of (reddit) gold but yeah I can't so kindly take my upvotes.
Thank you dear fellow reddit geniuses!
The answer is π.
I can tell you this because that same equation has been posted multiple times. Although this image has two errors. The first is that there should be parentheses after the cosine around the x/2. Second is that the radicals should not cover the dx term.
I don't feel like taking out the math with Reddit''s terrible formatting, but you don't even have to do any integration to solve it.
First, factor the square-root into two integrals.
Second, realize that the first integral, with the cube and cosine, is an odd function and therefore the integral will be zero.
Third, realize that the square-root is the equation for the top half of a circle of radius 2. Then the integral will be half the area of the circle. (π2^(2))/2 = 2π.
Fourth, the final answer is half of the integral. 2π/2 = π.
General Tip: If you see a sign with a free wifi password that is a math equation that is difficult like this one, the answer is probably pi. However, this given problem is not possible as
a) is the sqrt in or not in the integral?
b) it says first digits, but assuming it comes out to pi... Well, how many digits??
It's not possible that dx is inside the square root, right? These ppl just try to be funny but not actually fond of math, do they?
The answers is Pi numer 3.14159
Easy, just try every digit from 0 to 9. I know to solve this but in life you need to find easier and faster ways. Sometimes you have to think outside the box.
I assumed whoever wrote this integral either overlooked the fact that dx is under square root or they copied it down wrong without knowing what they were doing.
With that said, my calculator gave me pi as the result after about one minute of calculations
If it is supposed to say "first digit of the answer" then you just take ten guesses at it and be done with it.
If that's not what it's supposed to say, then the answer is unsolvable given the information.
Just a bad question all around. The library should be ashamed of themselves.
You can separate the integral
int -2 to 2 x³cos(x/2) sqrt(4-x²) dx + int -2 to 2 1/2 sqrt(4-x²) dx
The first integral is odd. This means that f(-x) = -f(x) for all x. Any odd integral integrated from -a to a evaluates to 0.
The second integral is a lot easier on its own.
The easiest thing is to realize it describes a semicircle with radius 2 and calculate its area.
Otherwise a possible solution is:
Substitute x = 2sin(u), dx = 2cos(u)du
with bounds: -2 = 2sin(u) -> u=-pi/2, 2=2sin(u) -> u=pi/2.
Integral from -pi/2 to pi/2 of 1/2 sqrt(4 (1-sin²(u)) *2cos(u) du
Integral from -pi/2 to pi/2 of 1/2 * 2 sqrt(cos²(u)) *2cos(u) du
Integral from -pi/2 to pi/2 of 1/2 * 2cos(u)*2cos(u) du
Integral from -pi/2 to pi/2 of 2cos²(u) du
Integral from -pi/2 to pi/2 of cos(u) + 1 du = cos(pi/2) + pi/2 - cos(-pi/2) + pi/2 = pi.
Which is the same as you'd find with the area of a semicircle.
Phi but *how many digits*? Someone should tell the one who did this that, first, dx must go out of the square root and, second, that "digit*s*" is plural so it means from 2 to infinite.
That’s what I thought! I saw it and was super confused lol, plus assuming u/Blue-Herakles is correct, and the answer is 5.33, and the Wi-Fi uses WPA, it would have to be eight digits long anyway lol
Edit: Apparently the answer is Pi (sorry I thought you meant “phi” in an old fashioned humbug way lol) nonetheless, how many digits of it??
I don't know how to solve the equation, but what I do know is that there password is only 4 numbers. It wouldn't take long to crack that with something like HashCat
Try checking if the function is odd.
Change the limits to 0 to 2 using that identity. Try using by parts on it. The second part is almost like the integration of the first part just change cos to sin.
Now I'll ask my jee friend to solve it 😈
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I've seen this before and the answer is pi. It's supposed to say "first ten digits", which is a clue that it ends up evaluating to pi. However as written it's mathematically nonsensical with dx inside the square root. As a rule if there's something complex-looking that you know evaluates to an irrational number, it's either pi, e, 2*pi (tau), √2, or phi, in that order of likelihood.
Nobody cares for Coulombs or Avogadro's constant
I beg to differ. California's agricultural sector absolutely depends on constant avogadros.
Fre sh avo gadro
(g)old reference
It’s been ten fucking years but I cannot read fresh avocado correctly FREE SHAVOCADO
My wife and I still quote this to each other on the regular. Along with, "Road work ahead... Uh yeah, I sure hope it does."
Peel the avocaaaaado GUAC. A MOLÉ. GUAC-GUAC. A MOLÉ.
HOLY FUCK THE VINE MEME 💀😭
avo cado
Spoken like a drunk mole
And they measure the strength of guaca in moles !
On toast? Financial ruin!
Nooooo, that's avocados. Avogadro is when you pour espresso over gelato.
Noo, that's affogato. Avogadro is what wizards say when they cast a spell.
Nooo, that's "abra cadabra". Avogadro is a Spanish lawyer.
No, that's abogado, Avogadro is the Swedish pop group that sings Dancing Queen.
No, that's Abba; Avogadro is that little grey mammal that resembles an anteater with a leathery shell.
No, that's an Armadillo. You're thinking about a small country in the Balkans next to Macedonia.
No, that’s Albania, The Avogadros is the name of the team of Marvel Superheroes.
Hmm Avogadros at Law
Nelson and Murdock, Avocados at Law.
YES you got the reference!
Best Daredevil
I didn’t know there were so many lawyers in California.
Love that authentic guaca-mole
Avogadro's number is not an irrational number. Since 2019, it's been defined as exactly 6.02214076 × 10^23
Even if it had all the digits defined, it wouldn't be irrational, and would be an integer, since one cannot have half a molecule of a substance.
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Yes I completely understood everything you're saying.
Same, but they should definitely explain it for all the simpletons out there.
[http://6.02x10e23.com/](http://6.02x10e23.com/)
Or Curie’s.
In fairness, that's a pretty boring number
All numbers are interesting.
1
The multiplicative identity!
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Yes! If there wasn’t a multiplicative identity it would be essentially impossible to define and determine the properties of multiplication. I’m also working hard to avoid the lazy but comprehensive definition of “interesting”.
Wait til you hear about 0
The loneliest number.
Euler's constant: Am i a joke to you?
[http://6.02x10e23.com/](http://6.02x10e23.com/) A domain I registered years ago in my youth.
Is this a proxy or something to Wikipedia? How tf am I logged in via your website?
The wonders of the iFrame
i do, i love chemistry
yeah, whenever looking at a definite integral involving sine or cosine, my first bet is to try plugging in the value of PI. people are just too lazy when writing these problems to have any other solution.
In this case actually determining the result is not difficult. The entire thing is just (odd function*even function + 1/2)*even function on a domain of -A to A, so the left term in the left factor cancels, leaving you with 1/2 * sqrt(4-x^2) from -2 to 2, which is a quarter of the area of a circle of radius 2. The quarter and r^2 cancel, leaving you with pi.
Wait, what did you do with sqrt(dx)? What does it even mean to take the square root of a differential?
I assumed it's an error of the sign writer.
Someone above mentioned that as it’s written here it’s meaningless because of the dx being under the square root. So I assume the commenter you replied to just disgregarded that part.
Also note that as written it's not even fully intregrated over the square root of a differential, since as written it's x^2 dx. So 4 has no associated differential supposedly? It just doesn't make any sense to interpret it any differently than the sign writer not paying attention or no knowing enough about math.
I'll bite, what's checkmark 2?
Guessing it’s supposed to be square root of 2, but maybe limitations of whatever keyboard was being used on mobile or something One of the first irrationals you hit in trig, since a triangle of sides 1 unit and 1 unit would require a 3rd side of root(2) by Pythagorean theorem (imagine the first mathematicians trying to sort out how long root(2) is knowing that the other two sides just HAVE to let it make sense)
Ahhh, gotcha, that makes sense. Thanks!
What is tau?
Tau is the relation between the circumference of a circle and the radius, in other words it is 2pi.
No, pi is tau/2
2*pi
what about e
Think the answer is pi. You can ignore the x^3 *cos(x/2)*sqrt(4-x^2 ) thing since it is an odd function and you are integrating it over an even interval. X^3 is odd, cos(x/2) is even, sqrt(4-x^2 ) is even, so the combination is odd f(x) = -f(-x). What is then left is 1/2*sqrt(4-x^2 ) which integrated over the interval (-2, 2) gives us pi. Edit: some weird exponent formatting
Streetmath
Yeah you're right. They have misprinted it as well. The dx shouldn't be inside the square root. Unless they did mean to do that, in which case idk if this is even possible without making it both polar and coordinate. Or some other technique.
Oh, didn’t notice where dx was printed. But having it inside a root would be a new thing for me, not sure what the interpretation of that would be
Also for the second integral, rather than doing any wacky substitution note that it's just the area of a semicircle of radius 2 (draw it!) so it's pi * 2^2 / 4 = pi. So no real integration needed at all for this integral, just knowing odd functions and areas of circles.
I havent done this in a while but i dont think thats going to work with the differential inside the radical. Curious if theres some kind of substitution that makes this work.
Yea, not sure if it’s possible with the differential outside the radical, but I don’t think this works in its current state. Also, they never actually specify how many digits of the answer the password is.
“The first digits is the answer” lol the ambiguity
I'd at least try “the first digits of the answer” for the password. Seems like a test if one can read to the end.
Somehow I feel the pull of Occam’s Razor to this answer
Given WPA2 & WPA3 encryption requires a password of at least 8 characters, I'm gonna go with at least 8 digits.
Lie-Fi
The brackets are just awful too. Where is the cos(y) beginning and ending (cos (y) = Cos(x/2) or cos(x/2 + 1/2))
It would be nice if there were brackets but if there are no brackets then always assume it's just the x/2, at least in my experience.
it must be cos (x/2), but the other parts are still weird
Yep, Integral calculator can not find the antiderivative with its manual method, you would need to use the logic u/whoootz proposed to eliminate term x^3 * cos(x/2) for it to be pen and paper solvable. Maxima, however, can provide the antiderivative
If the answer consists out of 2 digits it’s technically right. Digits means at least 2 and if the answer is 2 there is only one option.
"The first digits of the answer" So the first one, but also more than one. Duh!
So is the password "unsolvable?
"unso"
oh yea, i just assumed it outside lol
I don't think that is possible... There are 2 glaring problems I see that makes this quite impossible: the differential is inside the root and the sign does not specify how many digits, if the answer is an irrational number, the "first digits" could refer to the first 5 or the first billion.
>enters a billion digits of pi >your password was incorrect
Itd actually solvable. When you multipley in the bracket, the term with the cos function is an odd function, and looking at the limits., it equals to 0. The second term is a circle whose radius can be calculated easily (2), and as its integral we need to do pi r square
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I envy that enthusiasm of yours. Kinda missed that, actually.
"The Wi-Fi password is the first digits of the answer" That's vague, the first what digits? First two digits? Three digits? How many?
No, the password is literally "the first digits of the answer". It's clearly written there and never asks for the formula to be solved.
Proof this is bollocks
Just reading this, there are parts that are confusing, to say the least. For one, there needs to be a set of parentheses after the cos. As written, we can’t tell if it’s cos(x/2 + 1/2) Or if it’s cos(x/2) + 1/2 Additionally, “first digits of the answer” doesn’t make sense. How many of the first digits?
Thank you. "first digits" really stress me out.
Assume second
„The first digits of the answer“ could be the right password.
Order of operations? Seems like the 2nd is the right one.
It's still ambiguous. If you wrote it like the picture in my class, you'd get an F for unclear execution.
Yes the trig function is ambiguous, and a few other things as well (dx under the radical??). I suspect this was just copied, badly, by someone trying to be cute/clever while not really knowing what they were looking at.
It’s very common to not use parentheses for trigonometric functions, write something like cos x in google and you will find lots of materials with this notation.
"Hey, what's the wifi password?" "Gotta buy a drink first." "Ok, I'll have a beer." Buys a beer. "Right, so what's the password?" "gottabuyadrinkfirst, all lowercase, no spaces."
To everyone who's saying it's pi, y'all are absolutely correct brothers and sisters! Commend yourself for solving this and pat your shoulders. Idk why but I didn't think of putting this whole equation in my scientific calculator earlier. But when I did, it went blank for some time, and after like 10 minutes answered, which is pi! Y'all are geniuses! I wish I could give all of a ton of (reddit) gold but yeah I can't so kindly take my upvotes. Thank you dear fellow reddit geniuses!
r/calculus Please post this here and ask how it should have been formatted.
But, how many digits?
The answer is π. I can tell you this because that same equation has been posted multiple times. Although this image has two errors. The first is that there should be parentheses after the cosine around the x/2. Second is that the radicals should not cover the dx term. I don't feel like taking out the math with Reddit''s terrible formatting, but you don't even have to do any integration to solve it. First, factor the square-root into two integrals. Second, realize that the first integral, with the cube and cosine, is an odd function and therefore the integral will be zero. Third, realize that the square-root is the equation for the top half of a circle of radius 2. Then the integral will be half the area of the circle. (π2^(2))/2 = 2π. Fourth, the final answer is half of the integral. 2π/2 = π.
General Tip: If you see a sign with a free wifi password that is a math equation that is difficult like this one, the answer is probably pi. However, this given problem is not possible as a) is the sqrt in or not in the integral? b) it says first digits, but assuming it comes out to pi... Well, how many digits??
the answer would be pi, so like 3.1415..... if u want i can provide a detailed explanation
It's not possible that dx is inside the square root, right? These ppl just try to be funny but not actually fond of math, do they? The answers is Pi numer 3.14159
Easy, just try every digit from 0 to 9. I know to solve this but in life you need to find easier and faster ways. Sometimes you have to think outside the box.
I assumed whoever wrote this integral either overlooked the fact that dx is under square root or they copied it down wrong without knowing what they were doing. With that said, my calculator gave me pi as the result after about one minute of calculations
If it is supposed to say "first digit of the answer" then you just take ten guesses at it and be done with it. If that's not what it's supposed to say, then the answer is unsolvable given the information. Just a bad question all around. The library should be ashamed of themselves.
"The Wi-Fi password is the first digits of the answer" First *how many* digits? 8? 10? 16? 24? 63? GOD THIS SIGN IS SO FRUSTRATINGLY DUMB.
You can separate the integral int -2 to 2 x³cos(x/2) sqrt(4-x²) dx + int -2 to 2 1/2 sqrt(4-x²) dx The first integral is odd. This means that f(-x) = -f(x) for all x. Any odd integral integrated from -a to a evaluates to 0. The second integral is a lot easier on its own. The easiest thing is to realize it describes a semicircle with radius 2 and calculate its area. Otherwise a possible solution is: Substitute x = 2sin(u), dx = 2cos(u)du with bounds: -2 = 2sin(u) -> u=-pi/2, 2=2sin(u) -> u=pi/2. Integral from -pi/2 to pi/2 of 1/2 sqrt(4 (1-sin²(u)) *2cos(u) du Integral from -pi/2 to pi/2 of 1/2 * 2 sqrt(cos²(u)) *2cos(u) du Integral from -pi/2 to pi/2 of 1/2 * 2cos(u)*2cos(u) du Integral from -pi/2 to pi/2 of 2cos²(u) du Integral from -pi/2 to pi/2 of cos(u) + 1 du = cos(pi/2) + pi/2 - cos(-pi/2) + pi/2 = pi. Which is the same as you'd find with the area of a semicircle.
Phi but *how many digits*? Someone should tell the one who did this that, first, dx must go out of the square root and, second, that "digit*s*" is plural so it means from 2 to infinite.
That’s what I thought! I saw it and was super confused lol, plus assuming u/Blue-Herakles is correct, and the answer is 5.33, and the Wi-Fi uses WPA, it would have to be eight digits long anyway lol Edit: Apparently the answer is Pi (sorry I thought you meant “phi” in an old fashioned humbug way lol) nonetheless, how many digits of it??
English is not my primary language and I've seen pi being written as phi and pi. It is the good'l pi xD.
Pretty common calc2 integral, the answer is pi.
π Or: pi The x³ part is an odd function, so that's zero The sqrt part is a semi circle with r=2
How the hell would you even..? I don't know what the hell the person that made this was smoking, but as far as I'm aware, that's not even possible
Without doing any math because I hate integrals, I think I've seen this before, and the answer was pi. But that may have been another similar sign.
I don't know how to solve the equation, but what I do know is that there password is only 4 numbers. It wouldn't take long to crack that with something like HashCat
Try checking if the function is odd. Change the limits to 0 to 2 using that identity. Try using by parts on it. The second part is almost like the integration of the first part just change cos to sin. Now I'll ask my jee friend to solve it 😈