For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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33
votes
2answers
648 views

A strange integral

While browsing on Integral and Series, I found a strange integral posted by @sos440. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind because ...
33
votes
3answers
642 views

$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$

Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction. $$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
33
votes
2answers
460 views

Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$

Please help me to find a closed form for the infinite product $$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$ where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
33
votes
6answers
2k views

Why is there antagonism towards extended real numbers?

In my backstory, I was introduced to the geometric concept of infinity rather young, through reading about the inversive plane. In the course of learning calculus, I'm pretty sure I formed a concept ...
33
votes
2answers
2k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos ...
33
votes
2answers
1k views

Is there any connection between Green's Theorem and the Cauchy-Riemann equations?

Green's Theorem has the form: $$\oint P(x,y)dx = - \iint \frac{\partial P}{\partial x}dxdy , \oint Q(x,y)dy = \iint \frac{\partial Q}{\partial y}dxdy $$ The Cauchy-Riemann equations have the ...
33
votes
2answers
1k views

Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$

Regarding this problem, I conjectured that $$ I(r, s) = \int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx = 4 \pi ...
33
votes
1answer
846 views

Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$

I encountered this integral in my calculations: $$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm ...
32
votes
4answers
1k views

Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$?

Let $a$ be a non-zero real number. Is it true that $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ is independent on $a$ ? Any proof?
32
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 ...
32
votes
8answers
4k views

Why is $\int_{0}^{\infty} \frac {\ln x}{1+x^2} \mathrm{d}x =0$?

We had our final exam yesterday and one of the questions was to find out the value of: $$\int_{0}^{\infty} \frac {\ln x}{1+x^2} \mathrm{d}x $$ Interestingly enough, using the substitution ...
32
votes
4answers
1k views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
32
votes
1answer
679 views

Finding the sum- $x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$

If $S = x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$ Find S. Note:This is not a GP series.The powers are in GP. My Attempts so far: 1)If $S(x)=x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$ Then ...
31
votes
7answers
9k views

What function can be differentiated twice, but not 3 times?

In complex analysis class professor said that in complex analysis if a function is differentiable once, it can be differentiated infinite number of times. In real analysis there are cases where a ...
31
votes
6answers
2k views

Evaluating the integral, $\int_{0}^{\infty} \ln\left(1 - e^{-x}\right) \,\mathrm dx $

I recently got stuck on evaluating the following integral. I do not know an effective substitution to use. Could you please help me evaluate: $$\int_{0}^{\infty} \ln\left(1 - e^{-x}\right) \,\mathrm ...
31
votes
3answers
805 views

What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$?

/A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Worlfram alpha gives an indefinite form in terms of the logarithmic integral ...
31
votes
5answers
806 views

About a harmonic series problem : How to prove that $\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$

How to prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$$ $H_n$ is the n th harmonic number
31
votes
2answers
1k views

Integral $\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$$
31
votes
3answers
956 views

Limit with a big exponentiation tower

Find the value of the following limit: $$\huge\lim_{x\to\infty}e^{e^{e^{\biggl(x\,+\,e^{-\left(a+x+e^{\Large x}+e^{\Large e^x}\right)}\biggr)}}}-e^{e^{e^{x}}}$$ (original image) I don't ...
31
votes
6answers
2k views

Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis

I am studying for an exam and I have been studying my butt off during the winter break for it. During the course of my study I have written down quite a number of tricks, which in my opinion were ...
31
votes
3answers
2k views

Evaluating the infinite series $\sum\limits_{n=1}^\infty(\sin\frac1{2n}-\sin\frac1{2n+1})$

I've been bored and playing with infinite series and came across in my book the following problem, namely to determine the convergence of: $$ \sum_{n = 1}^{\infty} \left[\sin\left(1 \over 2n\right) - ...
30
votes
4answers
2k views

Is it necessary that every function is a derivative of some function?

I thought about this a lot and consulted a lot of people but everyone had contradicting answers. I am a high school student. please help.
30
votes
5answers
867 views

Evaluating $\int_0^1 x \tan(\pi x) \log(\sin(\pi x))dx$

What starting point would you recommend me for the one below? $$\int_0^1 x \tan(\pi x) \log(\sin(\pi x))dx $$ EDIT Thanks to Felix Marin, we know the integral evaluates to ...
30
votes
10answers
4k views

A really complicated calculus book

I've been studying math as a hobby, just for fun for years, and I had my goal to understand nearly every good undergraduate textbook and I think, I finally reached it. So now I need an another goal. ...
30
votes
3answers
1k views

Proof of Frullani's Theorem

How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously ...
30
votes
4answers
1k views

When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?

The question is: When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution? Edit I should ...
30
votes
2answers
868 views

Prove $\int_{0}^{\frac{\pi}{2}}\frac{dx}{1+\sin^2{(\tan{x})}}=\dfrac{\pi}{2\sqrt{2}}\left(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\right)$

Prove the following integral $$I=\int_{0}^{\frac{\pi}{2}}\dfrac{dx}{1+\sin^2{(\tan{x})}}=\dfrac{\pi}{2\sqrt{2}}\left(\dfrac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\right)$$ This integral result was ...
30
votes
1answer
833 views

Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$

How can I prove the following identity? $$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$$
30
votes
5answers
2k views

The Meaning of the Fundamental Theorem of Calculus

I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
30
votes
5answers
898 views

Evaluating $\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$

Some time ago I came across to the following integral: $$I=\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$$ What are the hints on how to compute this integral?
30
votes
4answers
718 views

Improper integral diverges

Let a real-valued function $f$ be continuous on $[0,1].$ Then there exists a number $a$ such that the integral $$\int_0^1\frac 1 {|f(x)-a|}\, dx $$ diverges. How to prove that statement?
30
votes
2answers
778 views

Are all limits solvable without L'Hospital Rule or Series Expansion

Is it always possible to find the limit of a function without using L'Hospital Rule or Series Expansion For example, $$\lim_{x\to0}\frac{\tan x-x}{x^3}$$ $$\lim_{x\to0}\frac{\sin x-x}{x^3}$$ ...
29
votes
4answers
4k views

Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$

It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why? I got: $2\mathrm dy\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\mathrm ...
29
votes
9answers
3k views

Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist?

Does $1.0000000000\cdots 1$ (with an infinite number of $0$ in it) exist?
29
votes
4answers
2k views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
29
votes
3answers
2k views

Integrate square of the log-sine integral: $\int_0^{\frac{\pi}{2}}\ln^{2}(\sin(x))dx$

$\displaystyle \int_{0}^{\frac{\pi}{2}} \ln(\sin(x))dx=-\frac{\pi}{2}\ln(2)$ is an integral that is common. But, how can we show ...
29
votes
5answers
2k views

$\ln|x|$ vs.$\ln(x)$? When is the $\ln$ antiderivative marked as an absolute value?

One of the answers to the problems I'm doing had straight lines: $$ \ln|y^2-25|$$ versus another problem's just now: $$ \ln(1+e^r) $$ I know this is probably to do with the absolute value. ...
29
votes
2answers
2k views

Closed form for $\int_0^1\sqrt{\frac{2-x}{(1-x)\,x}}\,\log\left(\frac{(2-x)\,x}{1-x}\right)dx$

This is somewhat similar to my previous question: Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$ Is it possible to find a closed form ...
29
votes
2answers
1k views

Integral $\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx$

I'm struggling with this integral $$I=\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx.\tag1$$ Mathematica could not evaluate it in a closed form. Its numeric value is ...
29
votes
4answers
1k views

Evaluation of $\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\,\mathrm{d}x$

I plan to evaluate $$\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\, \mathrm{d}x$$ and I need a starting point for both real and complex methods. Thanks ! Sis.
29
votes
2answers
500 views

Closed form for $\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}$

Here is another infinite sum I need you help with: $$\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}.$$ I was told it could be represented in terms of elementary functions and integers.
29
votes
4answers
1k views

tough integral involving $\sin(x^2)$ and $\sinh^2 (x)$

I ran across this integral I get no where with. Can someone suggest a method of attack?. $$\int_0^{\infty}\frac{\sin(\pi x^2)}{\sinh^2 (\pi x)}\mathrm dx=\frac{2-\sqrt{2}}{4}$$ I tried series, ...
29
votes
2answers
774 views

Prove that $\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$

Question. Let $$ f(x)=\!\left\{\,\,\, \begin{array}{ccc} \displaystyle{\left\lfloor{1\over x}\right\rfloor}^{-1}_{\hphantom{|_|}}&\text{if} & 0\lt x\le 1, \\ & \\ 0^{\hphantom{|^|}} ...
29
votes
2answers
881 views

A conjectural closed form for $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$

Let $$S=\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!},\tag1$$ its numeric value is approximately $S \approx 0.517977853388534047...$${}^{[more\ digits]}$ $S$ can be represented in terms of the ...
29
votes
3answers
906 views

$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$ Evaluate Integral

Evaluate $$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$$
29
votes
1answer
391 views

Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?

Are (integers) plus (elementary functions) plus (generalized hypergeometric functions) sufficient to represent any algebraic number? For example, the real algebraic number $\alpha\in(-1,0)$ ...
29
votes
3answers
706 views

Convergence of a series with repeated sines

Show that the series $$ \sum_{n=1}^\infty \frac{\sin\big(\sin(n)\big)}{n}, $$ converges. More generally, show that for every $k\in\mathbb N$ the series $$ \sum_{n=1}^\infty \frac{\sigma_k(n)}{n}, $$ ...
28
votes
20answers
3k views

Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?

As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself. thanks.
28
votes
4answers
859 views

Prove $\int_0^\infty \frac{\ln \tan^2 (ax)}{1+x^2}\,dx = \pi\ln \tanh(a)$

$$ \mbox{How would I prove}\quad \int_{0}^{\infty} {\ln\left(\,\tan^{2}\left(\, ax\,\right)\,\right) \over 1 + x^{2}}\,{\rm d}x =\pi \ln\left(\,\tanh\left(\,\left\vert\, ...