# Tagged Questions

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

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### List all degree one, two, three polynomials.

All degree one polynomials over $\mathbb{C}$ are of the form $ax+b$, where $a,b$ are complex numbers. How two write down all degree two (and three) polynomials over $\mathbb{C}$? I only need to write ...
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### An infinitely powered expression [duplicate]

Here's an expression I am struggling to evaluate: $$\LARGE {\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\:\cdot^{\:\cdot^{\:\cdot}}}}}}$$ The value turns out be $2$, but I don't understand how do we get it. Can ...
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### How to Prove the Chain Rule for Limits Using a $\varepsilon-\delta$ Argument?

I came across the chain rule for limits the other day and it interested me quite a bit and surprisingly I couldn't find the proof on the internet anywhere. From what I understand the chain rule for ...
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### Integral $\int_0^1\arctan(x)\arctan\left(x\sqrt3\right)\ln(x)dx$

I need to evaluate this integral: $$\int_0^1\arctan(x)\arctan\left(x\sqrt3\right)\ln(x)dx$$ Apparently, Maple and Mathematica cannot do anything with it, but I saw similar integrals to be evaluated in ...
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### Need to prove $\int_{0}^{1}\frac{x^{2n+1}}{\sqrt{1-x^{2}}}\;dx = \int_{0}^{\pi/2}\sin^{2n+1}\theta\;d\theta$ [closed]

Please help me. I'm having trouble trying to prove $\int_{0}^{1}\frac{x^{2n+1}}{\sqrt{1-x^{2}}}\;dx = \int_{0}^{\pi/2}\sin^{2n+1}\theta\;d\theta$
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### Convergence of a series $\sum {\left( {1 - \frac{{\sin {a_n}}}{{{a_n}}}} \right)}$

Given ${a_n}$ a converging series $\sum {{a_n}}$, and also that for every $n$, $a_n\ne0$. Does the series $\sum {\left( {1 - \frac{{\sin {a_n}}}{{{a_n}}}} \right)}$ converge? For every example I've ...
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### What's the $\otimes$-operator in the proof of Reynolds' transport theorem at Wikipedia?

In the proof of Reynolds' transport theorem at Wikipedia, they use the identities $$\nabla\cdot(v\otimes w)=v(\nabla\cdot w)+\nabla v\cdot w$$ and $$(a\otimes b)\cdot n=(b\cdot n)a\;,$$ where $n$ is ...
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### Find $\int_{0}^{\frac{\pi}{2}} \dfrac{\sin^n(x)}{\sin^n(x)+\cos^n(x)} dx$. [duplicate]

Find $\displaystyle \int_{0}^{\frac{\pi}{2}} \dfrac{\sin^n(x)}{\sin^n(x)+\cos^n(x)} dx$. I was told to switch the limits of integration then add them. How can I do that here?
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### write in form of $x+iy$

Can somebody help me, can somebody give a hint to solve this question $\dfrac{\log z}{z^{3}-(1+i)(z)}$ at $z=1+i$ actually, I want to but it in the form of $x+iy.$
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### Prove that $\sin(2\sin^{-1}(\alpha)) = 2\alpha \sqrt{-\alpha^2+1}$.

Prove that $\sin(2\sin^{-1}(\alpha)) = 2\alpha \sqrt{-\alpha^2+1}$. I was doing a trigonometric substitution problem in Calculus and came across this and wanted to know the proof of it.
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### Will integral be $\frac{\pi}{2}$?

Show that $\int \frac{1-cosx}{x^2}\ dx=\frac{\pi}{2}$. I used Taylor's series for cosx to find integral but I don't see intergal becoming equal to $\frac{\pi}{2}$ without any limits of integration. ...
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### $\lim_{x\to 0}\frac{\sin 3x+A\sin 2x+B\sin x}{x^5}$ without series expansion or L Hospital rule

If $$f(x)=\frac{\sin 3x+A\sin 2x+B\sin x}{x^5},$$ $x\neq 0$, is continuous at $x=0$, then find $A,B$ and $f(0)$. Do not use series expansion or L Hospital's rule. As $f(x)$ is continuous at $x=0$,...
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### Computing the Integral $\int \frac{\sqrt{x}}{x^2+x} dx$

Find $\displaystyle \int \dfrac{\sqrt{x}}{x^2+x} dx$. What would be the best way to integrate this? I saw the answer to this and it looked simple so that might mean the steps would be too?
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### Any sort of real world application for volumes of solids (revolution)$?$

Just wondering if you've ever encountered an actual situation that is related to the concept of integration of cross sectional area to find $3d$ volume.
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### Integrate $\int \frac{1}{x^4+4}dx$

Integrate $\displaystyle \int \dfrac{1}{x^4+4}dx$. I could try breaking this up into two quadratic trinomials, but that seems like it would be a lot of work. If that is the best way here how do I ...
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### Is there standard terminology to describe the not-quite-a-limit behavior of ${\tan( \log x) \over x}$ as x approaches infinity?

Suppose I want to describe the long term behavior of ${\tan(\log x) \over x}$ as x increases towards positive real infinity. Now, $$\lim_{x \rightarrow \infty}{\tan(\log x) \over x}$$ obviously ...
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### Limit of $\frac{1-2\cos(x)+\cos^2(2x)}{x^2}$

I tried to find the value of this limit without L'Hopital , but no luck $$\lim_{x\to 0}\frac{1-2\cos(x)+\cos^2(2x)}{x^2}$$
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### applications to calculus questions

A farmer has 80m length of fencing. He wants to use it to form 3 sides of a rectangular enclosure against an existing fence, which provides the 4th side. find the maximum area that he can enclose and ...
Consider the function $$E(z)=\int_{-\infty}^z\frac{e^t}{t}dt.\quad (1)$$ Substituting $t\mapsto -u$ one obtains $$E(z)=-\int_{-z}^{\infty}\frac{e^{-u}}{u}du\equiv Ei(z).\quad (2)$$ It is ...