A fun time — in mathematics. This one sophism is using integral and it is taken from Mudd Math Fun Facts — the calculation ends up with 0 = 1. The link is also gone now, as they revamped the website.
Sophism means a fallacious argument, especially one used deliberately to deceive.
Sophisticate originates from sophisticus (Medieval Latin), while sophism is from Greek sophisma — different origins, but lo! Sophism and sophisticate, both have sophis. And yes, sophism has its sophistication.
Quoted
Let's continue the last line:
$∫{1/x} dx = 1 + ∫{1/x} dx$
Subtract both sides with $∫{1/x} dx$.
$∫{1/x} dx - ∫{1/x} dx = 1 + ∫{1/x} dx - ∫{1/x} dx$
$0 = 1$ ⁉️🙋♀️🙋
Where did we get that 1 number on the right side all of the sudden?
The answer is at the first definition step. The u, dv, du, and v variables.
Those are not supposed to be that way.
If we have $∫{1/x} dx$ to be solved using integration by parts, then:
$∫udv = uv - ∫vdu$
$∫udv = ∫{1/x} dx$
Thus:
$u = 1$
$dv = {1/x} dx$
$du = 0$
$v = ∫{1/x} dx$
Plug those in:
$∫{1/x} dx = u.v - ∫v.du$
$∫{1/x} dx = 1.∫{1/x} dx - ∫ [∫{1/x} dx] . 0$
$∫{1/x} dx = 1.∫{1/x} dx - 0$
$∫{1/x} dx = ∫{1/x} dx$
Therefore:
$∫{1/x} dx = ∫{1/x} dx$
Subtract both sides with $∫{1/x} dx$:
$∫{1/x} dx - ∫{1/x} dx = ∫{1/x} dx - ∫{1/x} dx$
$0 = 0$
0 = 0. 👍What Is Wrong with That Defintion?
Well, back to that "proof", the steps will produce 0 = 1.
Therefore, integration by parts technique can't be applied to solve that. Well, it can be used, but... it will loop back to itself.
Properly Referenced
By looking at integral table, we will see this reference:
$∫ 1/x\ dx = ln\ |x| + C$
It is defined as that. In the table. 😶
Well, it's not literally a definition in the formal mathematical sense, it's more the final established result of a derivation that's been promoted to "reference entry" in an integral table.
In teaching and lookup contexts, it's treated like a definition because we just take it from the table without re-deriving it every time.
Thus we need to memorise that table. 🤦
We can always derive it, but then, we will also use a definition to get to that form. It started out from compound interest.
Compound interest is this bit:
$(1+1/n)^n$
And if we put extremely large value for $n$:
$(1+1/n)^n$ as $n→∞$
It will yield 2.71828... pattern.
Then there was Leonhard Euler to cement the e. It's the Euler's cheek.
e is 2.71828..., the mysterious number. It's under irrational numbers.
Irrational.
foo foo twaddle twaddle narf ∴ e
∴ means "therefore".
$ln$ is $log_e$ ➡️ "logarithm to base e".
It is Medieval Latin term. $l$ and $n$ ➡️ $ln$ ➡️ logarithmus naturalis.
It's "natural" because it is the Euler's cheek. Posterior, it naturally happens on a human (or other mammal), comes in pair.
Jest aside, it started from the bankers (back in 1600) — not bonkers — "curiosity". One sod perhaps said:
Hang on, what if we compound more often? Monthly, daily, hourly... infinitely often! Can we charge people for interest that grows continuously?
So they let $n → ∞$.
Surprise! Instead of exploding into chaos, that expression stabilises at a neat limit:
${lim}↙{n → ∞} (1+1/n)^n = e$
Thus, this form:
$∫ 1/x\ dx = ln\ |x| + C$
Is similar to:
$∫ 1/x\ dx = log_e\ |x| + C$
"Compound interest" but with huge n ➡️ "compound interest".
By the way, the 0 = 1 steps above are a reference fallacy, a self-referential definition disguised as a calculation. Improper use of integrals. 🧐
How to prove that reference bit?
It's a bladdy calculus definition.
As in,
(Teacher) That thing over there is called a chair.
(Pupil) Why?
(Teacher) Well, we name it that.
(Pupil) Why?
(Teacher) Because "hair" is taken.
(Pupil) Why?
(Teacher) 🤦 Because...🏃♂️➡️
(Teacher) (From afar, shouting.) Hahaha, this is called far! (Waving.)
(Pupil) 👀
But, here:
$∫ 1/x\ dx = ln\ |x| + C$
Is because the derivative of $ln|x|$ is $1/x$.
🤣
Once more:
If:
${d/{dx}}F(x) = f(x)$
Then:
$∫f(x) dx = F(x) + C$
In one frame:
${d/{dx}}F(x) = f(x)$ ⟺ $∫f(x) dx = F(x) + C$
Since:
$∫ 1/x\ dx = ln\ |x| + C$
So then (or, because):
${d/{dx}}ln\ |x| = 1/x$, for $x ≠ 0$
It's just swirling around. It's like:
That colour is red.
It's because:
Red is the colour of that.
Or this tautology:
The parent of Bob is Karen because Karen is Bob's parent.
Hm, well, yes.
When we go deeper:
$ln(x)$ := $∫_1^x {1/t} dt$
:= means "is defined as".
Ah! Yet, another definition. 😂
Here's another approach on Quora — the top voted answer is using e.
The Proof
Let's refine that parent-of-Bob tautology.
Integral chair is not banana, because the derivative of banana is not chair. 🤷
$∫$ chair $dx = F(x) + C ≠ ∫$ banana $dx$
Therefore, $0 ≠ 1$. 🤔
Have a splendid day! 😃
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