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

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39
votes
7answers
3k 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 ...
39
votes
2answers
2k 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 ...
38
votes
2answers
2k views

Evaluating $\int P(\sin x, \cos x) \text{d}x$

Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$. For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$. Is there a general method which allows us to evaluate the ...
38
votes
3answers
3k views

Why, although these functions have the same derivative, do they not differ by a constant?

I calculated the derivative of $\arctan\left(\frac{1+x}{1-x}\right)$ to be $\frac{1}{1+x^2}$. This is the same as $(\arctan)'$. Why is there no $c$ that satisfies $\arctan\left(\frac{1+x}{1-x}\right) ...
38
votes
5answers
818 views

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

Is there a closed form for the following infinite product? $$\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$$
38
votes
3answers
1k views

Making trigonometric substitutions rigorous

I've been tutoring some basic calculus, and it made me think about something pretty basic. Let me explain the problem by example: Say we are given the integral $\int \frac{x^2}{\sqrt{1-x^2}}\ ...
38
votes
2answers
1k views

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

I need the method to evaluate this integral (the closed-form if possible). $$ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx $$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
37
votes
3answers
1k views

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

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 ...
37
votes
3answers
494 views

Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?

I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it? Are there any ...
37
votes
3answers
3k views

Addition is to Integration as Multiplication is to ______

Addition is to Integration as Multiplication is to ______ ? Everyone knows that definite integration is "a way to sum continuum-many terms" in a rough sense. Can we "multiply continuum-many factors" ...
37
votes
3answers
770 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. $$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
37
votes
4answers
14k views

Are calculus and real analysis the same thing?

I guess this may seem stupid, but how calculus and real analysis are different from and related to each other? I tend to think they are the same because all I know is that the objects of both are ...
37
votes
3answers
1k views

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

Is it always possible to find the limit of a function without using L'Hôpital Rule or Series Expansion? For example, $$\lim_{x\to0}\frac{\tan x-x}{x^3}$$ $$\lim_{x\to0}\frac{\sin x-x}{x^3}$$ ...
37
votes
1answer
923 views

Integral $\int_0^1\frac{x^9\left(x^4+x^2-x-1-5\ln x\right)}{\left(x^{10}-1\right)\ln x}\mathrm dx$

A friend of mine sent me an integral that she had not been able to crack, and me neither. It comes with a result, but without a proof (I suppose it originated in some math contest). Could you please ...
37
votes
1answer
1k 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 ...
36
votes
4answers
5k 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 = ...
36
votes
8answers
5k views

Where is the flaw in this “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: ...
36
votes
16answers
4k views

Why does the derivative of sine only work for radians?

I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm ...
36
votes
6answers
3k views

Fun Geometric Series Puzzle

I recently was reminded of a puzzle I solved in college and thought I'd give it a shot again. However, being distanced from college math, I am having a harder time remembering how I arrived at the ...
36
votes
8answers
8k views

When to Stop Using L'Hôpital's Rule

I don't understand something about L'Hôpital's rule. In this case: $$ \begin{align} & {{}\phantom{=}}\lim_{x\to0}\frac{e^x-1-x^2}{x^4+x^3+x^2} \\[8pt] & ...
36
votes
6answers
1k views

A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$

Let $H_{n}$ be the nth harmonic number defined by $ H_{n} = \sum_{n=1}^{n} \frac{1}{k}$. I'm interested in knowing how to show that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}.$$ I tried ...
36
votes
2answers
2k views

Is it mathematically valid to separate variables in a differential equation?

I read the following statement in a book on Calculus, as part of my mathematics course: Technically this separation of $\frac{dy}{dx}$ is not mathematically valid. However, the resulting ...
36
votes
3answers
883 views

$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$

I need to find a closed-form for the following integral. Please give me some ideas how to approach it: $$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
36
votes
10answers
962 views

What are Different Approaches to Introduce the Elementary Functions?

Motivation We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the ...
36
votes
1answer
3k views

Convergence of $\sum \limits_{n=1}^{\infty}\sin(n^k)/n$

Does $S_k= \sum \limits_{n=1}^{\infty}\sin(n^k)/n$ converge for all $k>0$? Motivation: I recently learned that $S_1$ converges. I think $S_2$ converges by the integral test. Was the question ...
36
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 ...
36
votes
2answers
1k views

Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$

Here is another integral I'm trying to evaluate: $$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx.\tag1$$ A numeric approximation is: ...
36
votes
6answers
3k 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 ...
36
votes
5answers
1k views

A question on Taylor Series and polynomial

Suppose $ f(x)$ that is infinitely differentiable in $[a,b]$. For every $c\in[a,b] $ the series $\sum\limits_{n=0}^\infty \cfrac{f^{(n)}(c)}{n!}(x-c)^n $ is a polynomial. Is true that $f(x)$ is a ...
35
votes
7answers
11k 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 ...
35
votes
4answers
2k views

Find $\lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$

Find$$\lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}.$$ This is a recent exam question, which I couldn't figure out in the exam. My guess is it doesn't ...
35
votes
7answers
46k views

What is Jacobian Matrix?

What is Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone explain with examples? Thanks! :)
35
votes
4answers
2k views

How do I integrate the following? $\int{\frac{(1+x^{2})\mathrm dx}{(1-x^{2})\sqrt{1+x^{4}}}}$

$$\int{\frac{1+x^2}{(1-x^2)\sqrt{1+x^4}}}\mathrm dx$$ This was a Calc 2 problem for extra credit (we have done hyperbolic trig functions too, if that helps) and I didn't get it (don't think anyone ...
35
votes
4answers
1k 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 ...
35
votes
5answers
1k 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 ...
35
votes
5answers
740 views

Computing $\lim\limits_{n\to\infty} \Big(\sum\limits_{i = 1}^n \sum\limits_{j = 1}^n \frac1{i^2+j^2}-\frac{\pi}{2} \log(n)\Big)$.

In the chatroom we discussed about the asymptotic of $\displaystyle \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i^2+j^2}$, and if we think of the inverse tangent integral, it's easy to see that ...
35
votes
2answers
1k views

How to show that $\int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx = 0$

Evaluate the integral: $$ \int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx$$ The answer is $0,$ but I am unable to get it. There is some symmetry I can not see.
35
votes
2answers
915 views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
35
votes
2answers
531 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.
35
votes
2answers
2k 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 ...
35
votes
1answer
978 views

Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $

I've found here the following integral. $$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$ I've never seen it before and I also ...
35
votes
1answer
496 views

Is there an integral for $\pi^4-\frac{2143}{22}$?

In Ramanujan's Notebooks, Vol 4, p.48 (and a related one in Quarterly Journal of Mathematics, XLV, 1914) there are various approximations, including the close (by just $10^{-7}$), $$\pi^4 \approx ...
34
votes
21answers
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.
34
votes
4answers
2k views

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

I want to find a closed form for this integral: $$I=\int_0^1\frac{\ln^2x}{\sqrt{x^2-x+1}}dx\tag1$$ Mathematica and Maple cannot evaluate it directly, and I was not able to find it in tables. A numeric ...
34
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?
34
votes
13answers
43k views

What are the practical applications of the Taylor Series?

I started learning about the Taylor Series in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series. Question: ...
34
votes
5answers
6k views

How can a structure have infinite length and infinite surface area, but have finite volume?

Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis. One Dimension - Clearly this structure is infinitely long. Two Dimensions - Surface Area = ...
34
votes
7answers
3k 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 ...
34
votes
5answers
3k views

Which derivatives are eventually periodic?

What derivatives are eventually periodic? I have noticed that is $a_{n}=f^{(n)}(x)$, the sequence $a_{n}$ becomes eventually periodic for a multitude of $f(x)$. If $f(x)$ was a polynomial, and ...