8
votes
6answers
471 views

Can every indefinite integral of a discontinuous function be written in a way that “proves” something false?

I just saw the following fake proof. $$\int \frac1x dx =\int 1\cdot \frac1x dx=x\frac1x+\int x \frac1{x^2} dx = 1+ \int \frac1x dx$$ Which would imply $1=0$, hence the fake proof tag. The ...
5
votes
3answers
93 views

Why does this $u$-substitution zero out my integral?

Here's how I understand $u$-substitution working for an integral. Essentially, it involves substitution of differential expressions, allowing you to cancel out terms of the integrand. When we change ...
20
votes
4answers
2k views

Using Integration By Parts results in 0 = 1

I've run into a strange situation while trying to apply Integration By Parts, and I can't seem to come up with an explanation. I start with the following equation: $$\int \frac{1}{f} \frac{df}{dx} ...
5
votes
1answer
83 views

Why does the same limit work in one case but fail in another?

The following questions has been bugging me since high-school calculus. Please help me find my peace once and for all: Consider a revolution solid generated by rotating a nice curve $f(x)$ around the ...
4
votes
1answer
243 views

Problems with fake proofs of limit of sequences

I can hardly imagine an easier example of the fact that my understanding of the topic is more than rusty. I will divide the question in two parts to make the reading easier: 1) Background; 2) ...
2
votes
2answers
124 views

Fallacy applying Leibniz integral rule to problem of $x^2 = x+\ldots+x$ ($x$ times)

I was trying to provide the same answer my mathematics professor gave me when I asked to the problem raised in this thread. What I was told was that instead of summing what we were really doing is ...
1
vote
2answers
75 views

Proof that the derivative of a linear function is $0$.

We have a function $f(x)=x$ which is defined and is continuous on the set $S$ of all real numbers. The derivative at point $x$, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}h$$ Using the theorems of ...
-1
votes
1answer
87 views

Two equal to One? Is it correct?

Some people ask me that two, Two equal to One ...
11
votes
2answers
268 views

Indefinite integral. Where is the mistake?

The problem was to compute $I=\int x^2\sin^{-1}(x)\ dx$ (where $\sin^{-1}(x)$ is the inverse function of $\sin(x)$). The answer of my students: firstly, we put $u=\sin^{-1}(x)$, so $x=\sin(u)$ and ...
10
votes
1answer
503 views

What is wrong with this funny proof that 2 = 4 using infinite exponentiation?

Out of boredom, I decided to recall the following equation: $$x^{x^{x\cdots}} = 2.$$ Which, I simply rewrote like this: $x^2 = 2$, and therefore $x = \sqrt{2}$. Then I took a look at the more ...
5
votes
3answers
330 views

I am almost certain the book is wrong on this “proof” of a limit.

Advanced Mathematics by Mingming Chen, Zhengyou Guo Jingxian Yu, Jinqiu Li. Chemical Industry Press pg 28, section 1.4.2 Example 2. Prove $$\lim_{x \to 1} \frac{1}{x-1} = \infty$$ Proof ...
3
votes
2answers
140 views

How find $\Gamma{\left(\frac{8}{9}\right)}=\frac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$

show that $$\Gamma{\left(\dfrac{8}{9}\right)}=\dfrac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$$ where Gamma function:http://en.wikipedia.org/wiki/Gamma_function I found this problem is ...
11
votes
4answers
582 views

Does $\lim \frac {a_n}{b_n}=1$ imply $\lim \frac {f(a_n)}{f(b_n)}=1$?

I wanted to prove the seemingly simple statement: If $\lim \frac {a_n}{b_n}=1$ and $f$ continuous with $f(b_n)\neq0$ then $\lim \frac {f(a_n)}{f(b_n)}=1.$ I started promptly with \begin{align} ...
11
votes
2answers
446 views

What is wrong with this fake proof that $\lim\limits_{n\rightarrow \infty}\sqrt[n]{n!} = 1$?

$$\lim_{n\rightarrow \infty}\sqrt[n]{n!}=\lim_{n\rightarrow \infty}\sqrt[n]{1}*\sqrt[n]{2}\cdots\cdot\sqrt[n]{n}=1\cdot1\cdot\ldots\cdot1=1$$ I already know that this is incorrect but I am wondering ...
31
votes
12answers
2k views

explaining the derivative of $x^x$

You set the following exercise to your calculus class: Q1. Differentiate $y(x) = x^x$. A student submits the following solution: Let $g(a)=a^x$ and $f(x)=x$. Then $y(x) = g(f(x))$, so by ...
8
votes
4answers
309 views

What's wrong in this equation? [duplicate]

What's wrong in this equation? $$\underbrace{x+x+x+x+\cdots+x}_{x \textrm{ times}}=x^2$$ now differentiate w.r.t. 'x' both sides $$\underbrace{1+1+1+1+\cdots+1}_{x \textrm{ times}}=2x$$ So, $$x=2x$$ ...
1
vote
1answer
287 views

Funny Proof of $2=1$ [duplicate]

We know $x.x=x^2$(Consider $x\ne 0$) $x.x$ is adding $x$ $x$ times. So we have $x+x+\dots+x$($x$ times $x$ is added)=$x^2$......(1) Differentiating both sides of (1) we get, $1+1+\dots+1$(x ...
4
votes
2answers
201 views

Can someone please explain to me how I did this summation formula wrong?

I was trying to show that $\sum \limits_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$ but instead I got this $[\frac{n(n+1)}{2}]^2$ which from my understanding I basically proved another summation formula ...
6
votes
2answers
4k views

Problem when integrating $e^x / x$.

I made up some integrals to do for fun, and I had a real problem with this one. I've since found out that there's no solution in terms of elementary functions, but when I attempt to integrate it, I ...
22
votes
5answers
1k views

Trying to understand why circle area is not $2 \pi r^2$

I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below: The area of a square is like a line, the height (one dimension, length) ...
3
votes
2answers
549 views

How to disprove this fallacy that derivatives of $x^2$ and $x+x+x+\cdots\ (x\text{ times})$ are not same. [duplicate]

Possible Duplicate: Where is the flaw in this argument of a proof that 1=2? (Derivative of repeated addition) $$x^2=\underbrace{{x+x+x+\cdots+x}}_{x \text{ times}}$$ $$\therefore \frac ...
9
votes
1answer
260 views

Where is the flaw in evaluating the following integral?

I was trying to evaluate a complicated integral by substitution and along the way I got stuck in nonsensical answer. Surprisingly enough the point I wanted to discuss can be demonstrated using the ...
3
votes
4answers
302 views

A contradiction involving derivative [duplicate]

Possible Duplicate: Where is the flaw in this argument of a proof that 1=2? I am unable to find where the error is occurring in the following (I guess I can't take derivative, but why?): ...
33
votes
12answers
5k views

How to convince a layman that the $\pi = 4$ proof is wrong?

The infamous "$\pi = 4$" proof was already discussed here: Is value of $\pi$ = 4 ? And I have read all the answers, yet I think that they will not be of much help to me if I try to explain this ...
27
votes
6answers
3k views

Where is the flaw in this argument of a proof that 1=2? (Derivative of repeated addition)

Consider the following: $1 = 1^2$ $2 + 2 = 2^2$ $3 + 3 + 3 = 3^2$ Therefore, $\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$ Take the derivative of lhs and rhs and we get: ...